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Plancherel Measure and Asymptotic Young Diagrams

1 · Prerequisites

2 · Summary

This page studies the Plancherel measure on Young diagrams, the asymptotic shape of a typical diagram, and Kerov's central limit theorem for the normalized cycle characters, on the representation-theoretic base of frobenius-characteristic-and-the-symmetric-group-character-dictionary, specht-modules-and-the-irreducibles-of-the-symmetric-group, the-branching-rule-and-the-young-graph and the-hook-length-formula-and-rsk-correspondence, and the probability base of finite-probability-spaces-and-random-variables, modes-of-convergence-for-random-variables, weak-convergence-tightness-and-representation and central-limit-theorems and brownian-motion-construction-and-continuity (the Gaussian-moment supplier).

The measure itself is $P_{n} (λ) = (f^{λ})^{2} / n!$ (The Plancherel measure on the partitions of $n$ ), normalized by the sum-of-squares identity (The Plancherel weights sum to one) and realized as the law of the Robinson-Schensted shape of a uniform permutation (The RSK shape of a uniform random permutation has the Plancherel law). The scaled Russian profile of a diagram is introduced on Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram, its profile moments on Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ , and the limit profile $Ω$ on The Logan-Shepp-Vershik-Kerov limit profile $Ω$ , and the law of large numbers for the profile is proved first in moment form (Scaled Plancherel profile moments converge in probability) and then uniformly (Plancherel Young diagrams converge to the limit shape), using an elementary RSK union bound (The RSK union bound localizes Plancherel profiles) and the finite-moment topology of bounded Lipschitz profiles (Finitely many polynomial moments control the uniform distance on bounded Lipschitz profiles).

The character fluctuation theory is carried by the shifted character observables $p_{ρ}^{#}$ and their Hermite normalization. The algebra $A = R [{\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots]$ with the basis ${p_{ρ}^{#}}$ , the Kerov filtrations and the top-term multiplication rule are set up on Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ and The shifted character observables form a basis of $A$ , with the Kerov weight filtration, with the exact $p_{1}^{#}$ product and leading terms for $p_{k}^{#}$ in Shifted character products: exact for $p_{1}^{#}$ and leading terms for $p_{k}^{#}$ and the generator expansion The profile-moment generators in the shifted-character basis; the Plancherel expectations Plancherel expectations of the shifted character observables and the limit values The profile moments of $Ω$ are central binomial coefficients identify the multiplicative functional, and the monic Hermite polynomials (The monic probabilists' Hermite polynomials, Gaussian orthogonality and the monomial expansion of the Hermite polynomials, Hermite leading terms for normalized shifted characters) supply the moment comparison. Determinacy of the Gaussian limit is a local result (The standard Gaussian law is determined by its moments) feeding the multivariate moment method (The multivariate method of moments for a determinate limit), which proves Kerov's central limit theorem (Kerov's central limit theorem for normalized cycle characters) for joint convergence in distribution of the normalized cycle characters (Joint convergence in distribution and the normalized cycle-character observables, Normalized shifted character observables $η_{ρ}$ ). The closing remark The RSK and longest-increasing-subsequence consequences remain owned by the hook-length/RSK page records that no LIS fluctuation, Baik-Deift-Johansson or Tracy-Widom statement is claimed here. The Axiom of Choice is declared for the Gaussian target law, Gaussian determinacy, Hermite orthogonality, the multivariate moment method, and the character CLT. The Hermite recurrence itself, the finite measures, profiles and law-of-large-numbers arguments are choice-free.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The monic probabilists' Hermite polynomials

Definition

The monic probabilists' Hermite polynomials are the polynomials $H_{m} \in R [x]$ defined by $H_{0} = 1, H_{1} (x) = x, x H_{m} (x) = H_{m + 1} (x) + m H_{m - 1} (x) (m \geq 1) .$ The recurrence is solved for the higher polynomial, $H_{m + 1} (x) = x H_{m} (x) - m H_{m - 1} (x)$ , so it determines $H_{m}$ uniquely by induction on $m$ . Each $H_{m}$ is monic of degree $m$ : $H_{2} = x^{2} - 1$ , $H_{3} = x^{3} - 3 x$ and $H_{4} = x^{4} - 6 x^{2} + 3$ ; in general $H_{m} (x) = m! \sum_{j = 0}^{⌊ m / 2 ⌋} \frac{(- 1 / 2)^{j} x^{m - 2 j}}{j! (m - 2 j)!}$ .

Under the AC assumption of the Gaussian-law supplier, these are the monic orthogonal polynomials for the standard normal law of Standard normal and normal laws: they form an orthogonal system for the measure $(2 π)^{- 1 / 2} e^{- x^{2} / 2} d x$ ; orthogonality and the expansion of monomials in this system are proved separately on this page. The moments of that law are those of Moments, variance, and covariance on a probability space, and the polynomial calculus used below is that of The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set. The defining property used in this batch is the recurrence; no choice principle is used.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The Plancherel measure on the partitions of $n$

Definition

For an integer $n \geq 0$ let $Y_{n}$ be the set of partitions $λ ⊢ n$ of Partitions, English diagrams, and conjugation. This set is finite: a partition of $n$ has at most $n$ parts and every part is at most $n$ , so $Y_{n} \subseteq {(λ_{1}, \dots, λ_{k}) : k \leq n, n \geq λ_{1} \geq \dots \geq λ_{k} \geq 1}$ , a subset of the finite set ${1, \dots, n}^{\leq n}$ .

For $λ ⊢ n$ put $f^{λ} : = \dim_{C} S^{λ}$ , the dimension of the complex Specht module, so that $f^{λ}$ equals the number of standard $λ$ -tableaux (Standard polytabloids form a basis of a complex Specht module, The hook length formula); in particular $f^{λ}$ is a positive integer, because the standard polytabloids form a basis, and $f^{λ} = n! / \prod_{x \in [λ]} h (x)$ with the empty product $1$ for $λ = \emptyset$ , so that $f^{\emptyset} = 1$ .

The Plancherel measure of order $n$ is the function $P_{n} (λ) : = \frac{(f^{λ})^{2}}{n!}, λ \in Y_{n},$ with $n!$ the factorial of The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ . The conventions $0! = 1$ and $f^{\emptyset} = 1$ give $P_{0} (\emptyset) = 1 / 1 = 1$ . For every $λ$ the value $P_{n} (λ)$ is a well-defined nonnegative real number, being a quotient of a nonnegative integer by the positive integer $n!$ . Thus $P_{n}$ is a function on the finite set $Y_{n}$ . No choice principle is used: every quantity involved is finite. Normalization, $\sum_{λ ⊢ n} P_{n} (λ) = 1$ , is not part of this definition and is proved separately in The Plancherel weights sum to one.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Finitely many polynomial moments control the uniform distance on bounded Lipschitz profiles

Statement

Fix $I = [a, b]$ with $a \leq b$ and let $Σ_{I}$ be the set of real functions $σ$ with support in $I$ satisfying $∣ σ (x) - σ (y) ∣ \leq ∣ x - y ∣$ for all $x, y$ . Then for every $ε > 0$ there exist $K \in N$ and $δ > 0$ such that every $σ \in Σ_{I}$ with $∣ \int_{R} σ (x) x^{k} d x ∣ \leq δ (k = 0, 1, \dots, K)$ satisfies $\sup_{x \in R} ∣ σ (x) ∣ \leq ε$ . Consequently, if $(σ_{n}) \subseteq Σ_{I}$ and $\int_{R} σ_{n} (x) x^{k} d x \to 0$ for every $k \geq 0$ , then $σ_{n} \to 0$ uniformly on $R$ ; equivalently, on $Σ_{I}$ the topology of all polynomial moments coincides with the topology of uniform convergence.

Facts & Assumptions

Given: reals $a \leq b$ , the set $Σ_{I}$ of real functions $σ$ vanishing outside $I$ with $∣ σ (x) - σ (y) ∣ \leq ∣ x - y ∣$ for all $x, y$ , and a real $ε > 0$ . For $σ \in Σ_{I}$ the integral $\int_{R} σ (x) x^{k} d x$ of the Statement is read as $\int_{a}^{b} σ (x) x^{k} d x$ (Riemann-Darboux, The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ) when $a < b$ , and as $0$ when $a = b$ ; the convention $x^{0} = 1$ is used.

[F1]

A function with $∣ g (x) - g (y) ∣ \leq ∣ x - y ∣$ for all reals $x, y$ is continuous on $R$ : at every point and every real $η > 0$ , $δ : = η$ witnesses continuity (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

[F2]

For $a \leq b$ , every continuous real function on $[a, b]$ is a uniform limit of polynomials (Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval).

[F3]

For $a < b$ , every continuous real function on $[a, b]$ is bounded and Riemann integrable, so its Darboux integral exists (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

[F4]

For $a < b$ , if $f, g$ are integrable on $[a, b]$ then so are $f + g$ and $λ f$ for real $λ$ , with $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ); if $f \leq g$ pointwise on $[a, b]$ then $\int_{a}^{b} f \leq \int_{a}^{b} g$ , and if $m \leq f \leq M$ then $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[F5]

Absolute value and its basic inequalities: $∣ u ∣ = u$ for $u \geq 0$ and $∣ u ∣ = - u$ for $u < 0$ (Absolute value in an ordered field); for every real $c > 0$ and real $x$ , $∣ x ∣ \leq c$ if and only if $- c \leq x \leq c$ (Basic properties of the absolute value); and $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ for all reals $x, y$ (The triangle inequality).

[F6]

A sequence $(g_{n})$ of real functions converges uniformly to $0$ on $R$ when for every real $η > 0$ there is $N$ with $∣ g_{n} (x) ∣ < η$ for all $n \geq N$ and all $x$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

Proof

technique · direct

1.1givenF1algebra

Boundedness of the profiles: let $σ \in Σ_{I}$ . If $a = b$ , then for every real $r > 0$ the point $y : = a + r$ lies outside $I$ , so $σ (y) = 0$ and $∣ σ (a) ∣ \leq ∣ a - y ∣ = r$ ; hence $∣ σ (a) ∣ = 0$ and $σ \equiv 0$ . If $a < b$ , the Lipschitz bound together with $σ (y) = 0$ for $y < a$ gives $∣ σ (a) ∣ = ∣ σ (a) - σ (y) ∣ \leq a - y$ for every $y < a$ , hence $∣ σ (a) ∣ \leq 0$ , and symmetrically $σ (b) = 0$ ; then for $x \in I$ one has $∣ σ (x) ∣ \leq ∣ x - a ∣$ and $∣ σ (x) ∣ \leq ∣ b - x ∣$ , so $∣ σ (x) ∣ \leq (b - a) / 2$ . Put $C : = \max {1, (b - a) / 2}$ , so $\sup_{R} ∣ σ ∣ \leq C$ for every $σ \in Σ_{I}$ , and $σ$ is continuous on $R$ by [F1]. For the rest of the proof assume $a < b$ ; the case $a = b$ is finished below.

1.2givenF1F3F4F5constructalgebra

The comparison bump: fix $x \in I$ and a real $η > 0$ , and put $u : = \max (a, x - η / 2)$ , $v : = \min (b, x + η / 2)$ , so $u < v$ because $a < b$ and $x \in I$ . Let $c : = (u + v) / 2$ and define $F (y) : = \max {0, \frac{2}{v - u} (1 - \frac{2 ∣ y - c ∣}{v - u})}$ for real $y$ . Then $F \geq 0$ is continuous, its support is $[u, v] \subseteq I \cap [x - η / 2, x + η / 2]$ , and $\int_{R} F = 1$ , the graph of $F$ being a triangle of height $2 (v - u)^{- 1}$ and base $v - u$ . Since every $y$ in the support of $F$ satisfies $∣ y - x ∣ \leq η / 2$ , $σ (y) \geq σ (x) - ∣ y - x ∣ \geq σ (x) - η / 2$ ; hence if $σ (x) > η$ then $σ F \geq (η / 2) F$ pointwise on $I$ and, as $σ F$ and $F$ are continuous there, [F4] and [F3] give $\int_{R} σ F \geq (σ (x) - η / 2) \int_{R} F = σ (x) - η / 2 > η / 2$ , while if $σ (x) < - η$ then symmetrically $\int_{R} σ F \leq σ (x) + η / 2 < - η / 2$ . In either case $∣ σ (x) ∣ > η$ implies $∣ \int_{R} σ F ∣ > η / 2$ , so ${σ \in Σ_{I} : ∣ \int_{R} σ F ∣ \leq η / 2} \subseteq V (x, η) : = {σ \in Σ_{I} : ∣ σ (x) ∣ \leq η}$ .

1.3givenF1F3F4F5algebra

Integral triangle inequality and polynomial bounds: let $f$ be continuous on $[a, b]$ . Then $f$ and $∣ f ∣$ are continuous and integrable by [F3]. Applying [F4] to the two pointwise chains $- ∣ f ∣ \leq f \leq ∣ f ∣$ and $- f \leq ∣ f ∣$ gives $\int f \leq \int ∣ f ∣$ and $- \int f = \int (- f) \leq \int ∣ f ∣$ ; by [F5], $∣ \int f ∣ \leq \int ∣ f ∣$ . Now let $P = \sum_{k = 0}^{K} a_{k} x^{k}$ be a real polynomial with $K \geq 0$ , put $S : = 1 + \sum_{k = 0}^{K} ∣ a_{k} ∣ \geq 1$ , let $η > 0$ and set $δ_{0} : = η / (4 S)$ . If $σ \in Σ_{I}$ satisfies $∣ \int_{R} σ x^{k} ∣ \leq δ_{0}$ for $k = 0, \dots, K$ , then $σ P$ is continuous on $[a, b]$ , [F4] gives $\int_{R} σ P = \sum_{k = 0}^{K} a_{k} \int_{R} σ x^{k}$ , and the integral triangle inequality just proved together with [F5] yields $∣ \int_{R} σ P ∣ \leq \sum_{k = 0}^{K} ∣ a_{k} ∣ ∣ \int_{R} σ x^{k} ∣ \leq \sum_{k = 0}^{K} ∣ a_{k} ∣ δ_{0} \leq η / 4$ .

1.4givenchoosealgebra

A finite mesh: for every real $η > 0$ there are finitely many points $a = x_{1} < x_{2} < \dots < x_{m} = b$ with $x_{i + 1} - x_{i} \leq η$ for all $i < m$ ; one may take $m : = ⌈ (b - a) / η ⌉ + 1$ and split $[a, b]$ into equal parts. Every $x \in I$ then satisfies $∣ x - x_{i} ∣ \leq η$ for at least one mesh point $x_{i}$ .

2.1givenF1F2F3F4F5step 1.1step 1.2step 1.3algebra

Polynomial replacement of the bump: keep the notation of step 1.2 and put $θ : = η / (4 C (b - a)) > 0$ with $C$ from step 1.1. By [F2] choose a polynomial $P$ with $\sup_{y \in I} ∣ F (y) - P (y) ∣ \leq θ$ . The functions $σ F$ , $σ P$ and $σ (F - P)$ are continuous on $[a, b]$ by [F1], hence integrable by [F3], and all three vanish outside $I$ ; therefore, by step 1.3 and [F4], $∣ \int_{R} σ (F - P) ∣ \leq \int_{R} ∣ σ (F - P) ∣ \leq C (b - a) \sup_{I} ∣ F - P ∣ \leq η / 4$ , and if $∣ \int_{R} σ P ∣ \leq η / 4$ , then [F5] gives $∣ \int_{R} σ F ∣ \leq ∣ \int_{R} σ P ∣ + ∣ \int_{R} σ (F - P) ∣ \leq η / 2$ . Combined with step 1.2, ${σ \in Σ_{I} : ∣ \int_{R} σ P ∣ \leq η / 4} \subseteq V (x, η)$ .

2.2givenstep 1.4algebra

From mesh values to the supremum: let $η > 0$ , let $a = x_{1} < \dots < x_{m} = b$ be a mesh as in step 1.4, and let $σ \in Σ_{I}$ satisfy $∣ σ (x_{i}) ∣ \leq η$ for all $i$ . For $x \in I$ choose $i$ with $∣ x - x_{i} ∣ \leq η$ ; then $∣ σ (x) ∣ \leq ∣ σ (x_{i}) ∣ + ∣ x - x_{i} ∣ \leq 2 η$ , while $∣ σ (x) ∣ = 0 \leq 2 η$ for $x \notin I$ . Hence $\sup_{R} ∣ σ ∣ \leq 2 η$ .

3.1givenstep 1.1step 1.3step 1.4step 2.1step 2.2algebra

First claim: given $ε > 0$ , apply steps 2.1 and 1.3 with $η : = ε / 2$ at each mesh point $x_{1}, \dots, x_{m}$ of step 1.4: for each $i$ this produces a polynomial $P_{i} = \sum_{k = 0}^{K_{i}} a_{i, k} x^{k}$ such that ${σ \in Σ_{I} : ∣ \int_{R} σ P_{i} ∣ \leq ε / 8} \subseteq V (x_{i}, ε / 2)$ , and a threshold $δ_{i} : = \frac{ε / 8}{1 + \sum_{k = 0}^{K_{i}} ∣ a_{i, k} ∣} > 0$ such that the moments up to $K_{i}$ being at most $δ_{i}$ force $∣ \int_{R} σ P_{i} ∣ \leq ε / 8$ . Put $K : = \max_{i} K_{i}$ and $δ : = \min_{i} δ_{i} > 0$ ; both depend only on $I$ and $ε$ . Let $σ \in Σ_{I}$ satisfy $∣ \int_{R} σ x^{k} ∣ \leq δ$ for $k = 0, \dots, K$ . For each $i$ the moments up to $K_{i} \leq K$ are at most $δ \leq δ_{i}$ , so $∣ \int_{R} σ P_{i} ∣ \leq ε / 8$ and hence $∣ σ (x_{i}) ∣ \leq ε / 2$ ; step 2.2 with $η = ε / 2$ gives $\sup_{R} ∣ σ ∣ \leq ε$ . In the case $a = b$ every $σ \in Σ_{I}$ is $σ \equiv 0$ by step 1.1, so any $K$ and $δ$ work.

4.1givenF3F4F6step 1.1step 1.3step 3.1algebra∎

Consequence and topology: for a sequence with every moment tending to zero, apply step 3.1 with tolerance $ε / 2$ ; the finitely many moment conditions hold eventually, giving $\sup ∣ σ_{n} ∣ \leq ε / 2 < ε$ , hence uniform convergence by [F6]. To compare the topologies at an arbitrary $τ \in Σ_{I}$ , put $g : = (σ - τ) / 2 \in Σ_{I}$ . Given $ε > 0$ , step 3.1 at tolerance $ε / 4$ supplies $K, δ > 0$ ; if $∣ \int (σ - τ) x^{k} ∣ < 2 δ$ for $k \leq K$ , then $\sup ∣ σ - τ ∣ = 2 \sup ∣ g ∣ \leq ε / 2 < ε$ . Thus a finite intersection of moment neighborhoods of $τ$ lies in each uniform neighborhood. Conversely, for every $k$ , continuity and steps 1.3 and [F4] give $∣ \int (σ - τ) x^{k} ∣ \leq (b - a) \max {1, ∣ a ∣, ∣ b ∣}^{k} \sup ∣ σ - τ ∣$ , so each moment functional is continuous for the uniform topology. These two neighborhood containments prove equality of the topologies; when $a = b$ the space is the singleton zero profile by step 1.1.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The standard Gaussian law is determined by its moments

Statement

Assume AC. Let $X$ be a real random variable with $E ∣ X ∣^{k} < \infty$ for every $k \geq 0$ such that $E [X^{k}] = E [Z^{k}]$ for every $k \geq 0$ , where $Z \sim N (0, 1)$ (Standard normal and normal laws). Then $X \sim N (0, 1)$ . More generally, let $Z$ be a real random variable with moment generating function finite on a neighbourhood of $0$ , and set $m_{k} : = E [Z^{k}]$ . If $X$ has all absolute moments finite and $E [X^{k}] = m_{k}$ for every $k \geq 0$ , then $X$ has the same law as $Z$ .

Facts & Assumptions

Given: AC; real random variables $X$ and $Z$ with $E ∣ X ∣^{k} < \infty$ and $E ∣ Z ∣^{k} < \infty$ for every $k \geq 0$ , and $E [X^{k}] = E [Z^{k}]$ for every $k \geq 0$ . In the Gaussian case $Z \sim N (0, 1)$ . In the general case there is a real $r > 0$ with $E [e^{t Z}] < \infty$ for every real $t$ with $∣ t ∣ < r$ ; an "analytic moment generating function on a neighbourhood of $0$ " is read as exactly this finiteness assertion, which analyticity on an interval implies. Write $φ_{X}, φ_{Z}$ for the characteristic functions (Characteristic function of a real random variable) and $h : = φ_{X} - φ_{Z}$ .

[F1]

For every $j$ , if $E ∣ Y ∣^{j} < \infty$ then $φ_{Y} \in C^{j} (R)$ , $φ_{Y}^{(l)} (t) = E [(i Y)^{l} e^{i t Y}]$ for $0 \leq l \leq j$ , hence $∣ φ_{Y}^{(l)} (t) ∣ \leq E ∣ Y ∣^{l}$ and $φ_{Y}^{(l)} (0) = i^{l} E [Y^{l}]$ (Moments give derivatives of the characteristic function); the moments are those of Moments, variance, and covariance on a probability space.

[F2]

If $X, Y \in L^{2} (P)$ are real random variables then $E [∣ X Y ∣] \leq (E [X^{2}])^{1 / 2} (E [Y^{2}])^{1 / 2}$ (Cauchy-Schwarz for random variables).

[F3]

For $Z \sim N (0, 1)$ one has $E [Z^{2 m}] = (2 m - 1)!! = 1 \cdot 3 \dots (2 m - 1)$ for every integer $m \geq 1$ , and $E [Z^{2 m + 1}] = 0$ for every $m \geq 0$ by symmetry of the density $e^{- x^{2} / 2} / \sqrt{2 π}$ (Gaussian even moments for Brownian increments, Standard normal and normal laws).

[F4]

For $v \geq 0$ , $e^{v} = \sum_{j \geq 0} v^{j} / j!$ , so $e^{v} \geq v^{k} / k!$ for every integer $k \geq 0$ (The power-series, product-limit, IVP, functional-equation, and Picard definitions agree).

[F5]

Taylor remainder bound: if $f$ has derivatives through order $n + 1$ on the closed interval between $a$ and $x$ , with $∣ f^{(n + 1)} ∣ \leq M$ there, then $∣ R_{n, a} f (x) ∣ \leq M ∣ x - a ∣^{n + 1} / (n + 1)!$ , where $R_{n, a} f (x) = f (x) - T_{n, a} f (x)$ and $T_{n, a} f$ is the Taylor polynomial of degree at most $n$ (Taylor polynomials and their remainders, A uniform derivative bound gives a uniform Taylor remainder bound).

[F6]

Two Borel probability laws on $R$ with equal characteristic functions are equal (Uniqueness of a law from its characteristic function).

[F7]

For every real $x$ there is a natural number $n \geq 1$ with $x < n$ (Every complete ordered field is Archimedean).

[F8]

Expectations of integrable variables are linear, monotone for real variables, and satisfy $∣ E U ∣ \leq E ∣ U ∣$ (Linearity, monotonicity, and the modulus bound for expectation).

Proof

technique · direct

1.1givenF1algebraF8

Setup: by [F1], $φ_{X}$ and $φ_{Z}$ are $C^{\infty}$ on $R$ with $∣ φ_{Y}^{(l)} (t) ∣ \leq E ∣ Y ∣^{l}$ for every $l \geq 0$ and every real $t$ , and $φ_{Y}^{(l)} (0) = i^{l} E [Y^{l}]$ ; consequently $h = φ_{X} - φ_{Z}$ is $C^{\infty}$ with $h^{(l)} (0) = i^{l} (E [X^{l}] - E [Z^{l}]) = 0$ for every $l \geq 0$ , and $∣ h^{(l)} (t) ∣ \leq E ∣ X ∣^{l} + E ∣ Z ∣^{l}$ for all $t$ . Also, by the standing hypothesis, in the general case $A_{x} : = E [e^{x Z}] + E [e^{- x Z}] < \infty$ for every real $x$ with $0 < x < r$ .

1.2givenF2F3algebra

Gaussian moment bounds: let $m \geq 0$ . The arithmetic inequality $(2 m)! \leq 4^{m} (m!)^{2}$ holds for $m = 0$ and is preserved by passing from $m$ to $m + 1$ , since $(2 m + 2) (2 m + 1) \leq 4 (m + 1)^{2}$ ; hence for $m \geq 1$ , using [F3], $E [Z^{2 m}] = (2 m - 1)!! = (2 m)! / (2^{m} m!) \leq 4^{m} (m!)^{2} / (2^{m} m!) = 2^{m} m! \leq (\sqrt{2})^{2 m} (2 m)!$ , and for $m \geq 1$ the Cauchy-Schwarz bound [F2] gives $E ∣ Z ∣^{2 m - 1} \leq (E [Z^{4 m - 2}])^{1 / 2} = ((4 m - 3)!!)^{1 / 2} \leq (2^{2 m - 1} (2 m - 1)!)^{1 / 2} \leq (\sqrt{2})^{2 m - 1} (2 m - 1)!$ , while $E ∣ Z ∣^{0} = 1$ . Therefore $E ∣ Z ∣^{k} \leq (\sqrt{2})^{k} k!$ for every $k \geq 0$ ; and in the Gaussian case $E ∣ X ∣^{k} \leq (E [X^{2 k}])^{1 / 2} = (E [Z^{2 k}])^{1 / 2} = ((2 k - 1)!!)^{1 / 2} \leq 2^{k / 2} (k!)^{1 / 2} \leq (\sqrt{2})^{k} k!$ for $k \geq 1$ , with equality $E ∣ X ∣^{0} = 1$ for $k = 0$ . Thus $E ∣ Y ∣^{k} \leq 4^{k} k!$ for $Y \in {X, Z}$ and every $k \geq 0$ , in the Gaussian case.

1.3givenF5algebra

Local vanishing: let $g$ be a real-valued $C^{\infty}$ function on $R$ and suppose there are $M > 0$ , $τ > 0$ with $∣ g^{(l)} (t) ∣ \leq M l! τ^{- l}$ for all $l \geq 0$ and all real $t$ , and let $t_{0}$ be a point with $g^{(l)} (t_{0}) = 0$ for every $l \geq 0$ . Then for every real $t$ with $∣ t - t_{0} ∣ < τ$ and every $j \geq 0$ , the Taylor polynomial satisfies $T_{j, t_{0}} g (t) = 0$ , so [F5] applied with $n = j$ and the bound on the $(j + 1)$ -th derivative gives $∣ g (t) ∣ = ∣ R_{j, t_{0}} g (t) ∣ \leq M (j + 1)! τ^{- (j + 1)} ∣ t - t_{0} ∣^{j + 1} / (j + 1)! = M (∣ t - t_{0} ∣ / τ)^{j + 1}$ ; letting $j \to \infty$ gives $g (t) = 0$ . Hence $g$ vanishes on $(t_{0} - τ, t_{0} + τ)$ .

2.1givenF2F4step 1.2algebraF8

MGF moment bounds: fix $0 < x < r$ and put $A : = E [e^{x Z}] + E [e^{- x Z}] \geq 2$ . Since $e^{x ∣ Z ∣} \leq e^{x Z} + e^{- x Z}$ , [F4] gives $E ∣ Z ∣^{k} \leq A k! x^{- k}$ for every $k \geq 0$ , including $k = 0$ . By [F2] and moment equality, $E ∣ X ∣^{k} \leq (E [X^{2 k}])^{1 / 2} = (E [Z^{2 k}])^{1 / 2} \leq A^{1 / 2} \sqrt{(2 k)!} x^{- k} \leq A^{1 / 2} 2^{k} k! x^{- k}$ , using the factorial inequality proved in step 1.2. Hence $E ∣ Y ∣^{k} \leq C k! (x / 2)^{- k}$ for $Y \in {X, Z}$ , with $C : = \max {A, A^{1 / 2}}$ . No integral over a zero power is used.

2.2givenF7step 1.3algebra

Global vanishing: let $g$ be as in step 1.3 and suppose in addition that $g^{(l)} (0) = 0$ for every $l \geq 0$ . Then $g \equiv 0$ on $R$ : step 1.3 with $t_{0} = 0$ gives $g \equiv 0$ on $I_{0} : = (- τ, τ)$ ; suppose $g \equiv 0$ on $I_{j} : = (- (1 + j / 2) τ, (1 + j / 2) τ)$ for some $j \geq 0$ and put $t_{\pm} : = \pm (j + 1) τ / 2$ , so $∣ t_{\pm} ∣ = (1 + j / 2) τ - τ / 2$ lies in the interior of $I_{j}$ and all derivatives of $g$ vanish at $t_{\pm}$ . Step 1.3 at $t_{\pm}$ gives $g \equiv 0$ on $((j - 1) τ / 2, (j + 3) τ / 2)$ and on $(- (j + 3) τ / 2, - (j - 1) τ / 2)$ ; since $(j + 3) τ / 2 = (1 + (j + 1) / 2) τ$ and $(j - 1) τ / 2 < (1 + j / 2) τ$ , the union of these intervals with $I_{j}$ contains $I_{j + 1} : = (- (1 + (j + 1) / 2) τ, (1 + (j + 1) / 2) τ)$ . By induction $g \equiv 0$ on $I_{j}$ for every $j \geq 0$ ; for an arbitrary real $T$ , [F7] supplies a natural number $n \geq 1$ with $n > 2 ∣ T ∣ / τ$ , hence $∣ T ∣ < n τ / 2 < (1 + (n - 1) / 2) τ$ and $T \in I_{n - 1}$ , so $g (T) = 0$ .

3.1givenF6step 1.1step 1.2step 2.2algebra

Gaussian case: by step 1.1, $h^{(l)} (0) = 0$ for every $l$ , and by steps 1.2 and 1.1, $∣ h^{(l)} (t) ∣ \leq E ∣ X ∣^{l} + E ∣ Z ∣^{l} \leq 2 \cdot 4^{l} l! = 2 l! (1 / 4)^{- l}$ for all $t, l$ . Thus both $Re h$ and $Im h$ satisfy the real Taylor hypotheses of step 2.2 with $M = 2$ and $τ = 1 / 4$ ; applying it separately to the two components gives $h \equiv 0$ , that is $φ_{X} = φ_{Z}$ . By [F6] the laws of $X$ and $Z$ are equal, so $X \sim N (0, 1)$ .

3.2givenF6step 1.1step 2.1step 2.2algebra

General case: fix $0 < x < r$ and let $C, τ : = x / 2$ be as in step 2.1, so $E ∣ Y ∣^{k} \leq C k! τ^{- k}$ for $Y \in {X, Z}$ and every $k \geq 0$ . By steps 1.1 and 2.1, $h^{(l)} (0) = 0$ for every $l$ and $∣ h^{(l)} (t) ∣ \leq 2 C l! τ^{- l}$ for all $t, l$ ; step 2.2 with $M = 2 C$ , applied separately to $Re h$ and $Im h$ , gives $h \equiv 0$ , hence $φ_{X} = φ_{Z}$ , and [F6] gives that $X$ and $Z$ have the same law.

4.1givenstep 3.1step 3.2∎

Conclusion: if $X$ has all moments and $E [X^{k}] = E [Z^{k}]$ for all $k$ with $Z \sim N (0, 1)$ , step 3.1 shows $X \sim N (0, 1)$ , which is the first assertion; if $Z$ has an analytic moment generating function on a neighbourhood of $0$ (so that the finiteness hypothesis of step 2.1 holds) and $X$ has all moments with $E [X^{k}] = m_{k}$ , step 3.2 shows that $X$ has the same law as $Z$ , which is the general assertion.

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Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram

Definition

A continual diagram is a function $ω : R \to R$ such that $∣ ω (x) - ω (y) ∣ \leq ∣ x - y ∣$ for all real $x, y$ (the Lipschitz condition) and $ω (x) = ∣ x ∣$ for all sufficiently large $∣ x ∣$ ; the set of continual diagrams is denoted $D_{0}$ . For $ω \in D_{0}$ set $σ_{ω} (x) : = \frac{1}{2} (ω (x) - ∣ x ∣) .$ Since $ω$ agrees with $∣ \cdot ∣$ outside a compact set, $σ_{ω}$ is compactly supported; and $σ_{ω}$ is 1-Lipschitz, because $∣ σ_{ω} (x) - σ_{ω} (y) ∣ = \frac{1}{2} ∣ ω (x) - ω (y) - (∣ x ∣ - ∣ y ∣) ∣ \leq \frac{1}{2} (∣ x - y ∣ + ∣ x - y ∣)$ by the Lipschitz bound and $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ : the latter follows by applying The triangle inequality to $x = (x - y) + y$ and to $y = (y - x) + x$ , with Basic properties of the absolute value. For $s > 0$ define the $s$ -scaling $ω_{s} (x) : = s^{- 1} ω (s x)$ . Then $ω_{s} \in D_{0}$ , and $σ_{ω_{s}} (x) = s^{- 1} σ_{ω} (s x), x \in R,$ directly from the definition, so scaling preserves the Lipschitz constant.

The empty partition has profile $\emptyset (x) : = ∣ x ∣$ . Every nonempty $λ ⊢ n$ determines a continual diagram $λ (\cdot) \in D_{0}$ (Partitions, English diagrams, and conjugation). Regard the boxes of $[λ]$ as the unit squares $[i - 1, i] \times [j - 1, j]$ , $1 \leq i \leq k$ , $1 \leq j \leq λ_{i}$ , in the plane with coordinates $(r, s)$ , and rotate by $x = s - r$ , $y = r + s$ . The outer staircase of the image, extended by the two axis rays, is the graph of a continuous piecewise linear function $y = λ (x)$ , with $λ^{'} (x) = \pm 1$ at every point where the derivative exists (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set): each horizontal or vertical unit step of the boundary staircase becomes a unit step of slope $\mp 1$ under the linear map $(r, s) \mapsto (s - r, r + s)$ , and the graph is read from the outer corner at $x = - λ_{1}^{'}$ to the corner at $x = λ_{1}$ . Outside these corners the boundary follows the axis strip, so $λ (x) = ∣ x ∣ for x \geq λ_{1} and for x \leq - λ_{1}^{'},$ the extreme corners being the end of the first row and the bottom of the first column. Hence $λ (\cdot) \in D_{0}$ and the support of $σ_{λ}$ is contained in $[- λ_{1}^{'}, λ_{1}]$ . The area identity, also valid for the empty profile, holds: $\frac{1}{2} \int_{R} (λ (x) - ∣ x ∣) d x = n,$ because the compact region ${(x, y) : - λ'_{1} \leq x \leq λ_{1}, ∣ x ∣ \leq y \leq λ (x)}$ is exactly the image under the invertible linear map $T (r, s) : = (s - r, r + s)$ , of determinant $- 2$ , of the union $[λ]$ of the $n$ unit squares, and the image of a Jordan-measurable compact set of area $n$ has area $∣ \det T ∣ \cdot n = 2 n$ (Change of variables for an injective $C^{1}$ map on a compact Jordan set, applied with $f \equiv 1$ ); the region is the area under the continuous piecewise linear function $λ (x) - ∣ x ∣ \geq 0$ over the compact interval $[- λ_{1}^{'}, λ_{1}]$ , which is its Riemann-Darboux integral (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ).

The $\sqrt{n}$ -scaled profile of $λ ⊢ n$ , $n \geq 1$ , is $\overset{ˉ}{λ} (x) : = n^{- 1 / 2} λ (n^{1 / 2} x) = λ_{n^{1 / 2}} (x), x \in R,$ the $\sqrt{n}$ -scaling of the preceding paragraph. Thus $\overset{ˉ}{λ} \in D_{0}$ , $\overset{ˉ}{λ} (x) = ∣ x ∣$ for $∣ x ∣ \geq \max (λ_{1}, λ_{1}^{'}) / \sqrt{n}$ , and $σ_{\overset{ˉ}{λ}} (x) = n^{- 1 / 2} σ_{λ} (n^{1 / 2} x)$ ; the area identity scales to $\frac{1}{2} \int_{R} (\overset{ˉ}{λ} (x) - ∣ x ∣) d x = 1$ . The probability distribution with which $λ$ is drawn in this batch is the Plancherel measure of The Plancherel measure on the partitions of $n$ . No choice principle is used.

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Gaussian orthogonality and the monomial expansion of the Hermite polynomials

Statement

Assume AC, and let $Z \sim N (0, 1)$ (Standard normal and normal laws). For the monic Hermite polynomials $H_{m}$ of The monic probabilists' Hermite polynomials:

(i) $E [H_{m} (Z)] = 0$ for every $m \geq 1$ , and $E [H_{m} (Z) H_{n} (Z)] = m! δ_{m n}$ for all $m, n \geq 0$ ;

(ii) for every $m \geq 0$ , $x^{m} = \sum_{j = 0}^{⌊ m / 2 ⌋} c_{m, j} H_{m - 2 j} (x)$ with $c_{m, 0} = 1$ and $c_{m, j} = \frac{m!}{2^{j} j! (m - 2 j)!} = (\binom{m}{2 j}) (2 j - 1)!! \in Z (0 \leq j \leq ⌊ m / 2 ⌋);$ equivalently, the monomials and the Hermite polynomials are related by a unitriangular change of basis in each finite degree;

(iii) consequently, for any $N \geq 1$ and any $m_{1}, \dots, m_{N} \in N$ , the mixed monomial $\prod_{k = 1}^{N} x_{k}^{m_{k}}$ is a $Z$ -linear combination of products $\prod_{k} H_{m_{k}^{'}} (x_{k})$ with $m_{k}^{'} \leq m_{k}$ and $m_{k}^{'} \equiv m_{k} (m o d 2)$ , the coefficient of $\prod_{k} H_{m_{k}}$ being $1$ .

Facts & Assumptions

Given: AC; a random variable $Z \sim N (0, 1)$ ; the polynomials $H_{m}$ defined by $H_{0} = 1$ , $H_{1} (x) = x$ and $x H_{m} = H_{m + 1} + m H_{m - 1}$ (The monic probabilists' Hermite polynomials); $φ_{Z} (t) = E [e^{i t Z}]$ is the characteristic function of $Z$ .

[F1]

For $Z \sim N (0, 1)$ and every real $t$ , $φ_{Z} (t) = e^{- t^{2} / 2}$ (Characteristic function of a normal law).

[F2]

If $E ∣ Y ∣^{k} < \infty$ then $φ_{Y} \in C^{k} (R)$ with $φ_{Y}^{(j)} (t) = E [(i Y)^{j} e^{i t Y}]$ for $0 \leq j \leq k$ ; in particular $φ_{Y}^{(j)} (0) = i^{j} E [Y^{j}]$ (Moments give derivatives of the characteristic function).

[F3]

For $Z \sim N (0, 1)$ , $E ∣ Z ∣^{2 m} = (2 m - 1)!!$ for every $m \geq 1$ (Gaussian even moments for Brownian increments); and $E ∣ X Y ∣ \leq (E X^{2})^{1 / 2} (E Y^{2})^{1 / 2}$ for square-integrable $X, Y$ (Cauchy-Schwarz for random variables).

[F4]

Derivative rules: if $g$ is differentiable then $t \mapsto g (- t)$ has derivative $- g^{'} (- t)$ , and $(f g)^{'} = f^{'} g + f g^{'}$ , $(α f)^{'} = α f^{'}$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F5]

Factorials: $m!$ is the product of $1, \dots, m$ with $0! = 1$ , and $(\binom{m}{2 j}) = \frac{m!}{(2 j)! (m - 2 j)!}$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ).

[F6]

Expectations of integrable variables are linear, monotone for real variables, and satisfy $∣ E U ∣ \leq E ∣ U ∣$ (Linearity, monotonicity, and the modulus bound for expectation).

Proof

technique · direct

1.1givenF1F2F3F4algebra

Vanishing of odd moments: by [F1] the function $φ_{Z} (t) = e^{- t^{2} / 2}$ is even, and by induction with [F4] each derivative $φ_{Z}^{(j)}$ has parity $(- 1)^{j}$ (differentiating flips parity); hence $φ_{Z}^{(2 k + 1)}$ is odd and therefore $φ_{Z}^{(2 k + 1)} (0) = 0$ for every $k \geq 0$ . All absolute moments of $Z$ are finite, since by [F3] $E ∣ Z ∣^{2 m} = (2 m - 1)!!$ and [F3] gives $E ∣ Z ∣^{2 m + 1} \leq (E ∣ Z ∣^{4 m + 2})^{1 / 2} < \infty$ ; so [F2] applies to every order and gives $E [Z^{2 k + 1}] = i^{- (2 k + 1)} φ_{Z}^{(2 k + 1)} (0) = 0$ .

1.2givenF4algebra

Derivative relation: $H_{m}^{'} = m H_{m - 1}$ for every $m \geq 1$ . This holds for $m = 1$ , since $H_{1}^{'} = 1 = H_{0}$ , and for $m = 2$ , since $H_{2}^{'} = 2 x = 2 H_{1}$ ; for $m \geq 2$ , if it holds for all indices up to $m$ , then differentiating $H_{m + 1} = x H_{m} - m H_{m - 1}$ with [F4] gives $H_{m + 1}^{'} = H_{m} + x H_{m}^{'} - m H_{m - 1}^{'} = H_{m} + m x H_{m - 1} - m (m - 1) H_{m - 2}$ , and substituting $x H_{m - 1} = H_{m} + (m - 1) H_{m - 2}$ yields $H_{m + 1}^{'} = (m + 1) H_{m}$ .

1.3givenF5algebra

Monomial expansion: every $m \geq 0$ admits the expansion $x^{m} = \sum_{j = 0}^{⌊ m / 2 ⌋} c_{m, j} H_{m - 2 j}$ with $c_{m, j} = \frac{m!}{2^{j} j! (m - 2 j)!}$ . Indeed $x^{0} = H_{0}$ gives $m = 0$ ; if the expansion holds for $m - 1 \geq 0$ , then multiplying by $x$ and using $x H_{l} = H_{l + 1} + l H_{l - 1}$ gives coefficient of $H_{m - 2 j}$ equal to $c_{m - 1, j} + (m + 1 - 2 j) c_{m - 1, j - 1}$ (with $c_{r, l} : = 0$ whenever $l < 0$ or $2 l > r$ , and $x H_{0} = H_{1}$ supplying the boundary case), which equals $c_{m, j}$ : for $j \geq 1$ the common denominator $2^{j} j! (m - 2 j)!$ turns it into $\frac{(m - 1)! (m - 2 j) + 2 j (m - 1)!}{2^{j} j! (m - 2 j)!} = \frac{m!}{2^{j} j! (m - 2 j)!}$ (using [F5]), and for $j = 0$ it gives $c_{m - 1, 0} = 1 = c_{m, 0}$ . By [F5], $c_{m, j} = (\binom{m}{2 j}) (2 j - 1)!!$ is an integer and $c_{m, 0} = 1$ . Moreover the monicity and degree $\deg H_{l} = l$ make the matrix of coefficients of $x^{0}, \dots, x^{m}$ in the basis $H_{0}, \dots, H_{m}$ unitriangular with diagonal entries $c_{m, 0} = 1$ , so the expansion is the unique one and defines an invertible unitriangular change of basis in each finite degree.

2.1givenF1F3step 1.1algebraF6

Stein identity: for every real polynomial $p$ , $E [Z p (Z)] = E [p^{'} (Z)]$ . Write $p (x) = \sum_{j = 0}^{d} a_{j} x^{j}$ ; then $E [Z p (Z)] = \sum_{j = 0}^{d} a_{j} E [Z^{j + 1}]$ and $E [p^{'} (Z)] = \sum_{j = 1}^{d} j a_{j} E [Z^{j - 1}]$ , both finite sums. The constant term contributes $a_{0} E [Z] = 0$ on the left and nothing on the right, and for every $j \geq 1$ one has $E [Z^{j + 1}] = j E [Z^{j - 1}]$ : when $j$ is even both sides vanish by step 1.1; when $j$ is odd, [F3] gives $E [Z^{j + 1}] = (j)!!$ and $j E [Z^{j - 1}] = j (j - 2)!! = j!!$ ; here $(- 1)!! : = 1$ covers $j = 1$ , where both sides are $E [Z^{2}] = 1$ . Hence the two sums are equal.

2.2givenstep 1.3algebra

Multivariate expansion: let $N \geq 1$ and $m_{1}, \dots, m_{N} \geq 0$ . Expanding each factor by step 1.3 and multiplying out, $\prod_{k} x_{k}^{m_{k}} = \sum_{j_{1}, \dots, j_{N}} (\prod_{k} c_{m_{k}, j_{k}}) \prod_{k} H_{m_{k} - 2 j_{k}} (x_{k})$ ; each index $m_{k}^{'} : = m_{k} - 2 j_{k}$ satisfies $m_{k}^{'} \leq m_{k}$ and $m_{k}^{'} \equiv m_{k} (m o d 2)$ , the coefficients are integers by step 1.3, and the single tuple $(j_{1}, \dots, j_{N}) = (0, \dots, 0)$ contributes $\prod_{k} H_{m_{k}}$ with coefficient $1$ .

3.1givenstep 1.1step 1.2step 2.1algebraF6

Zero means: $E [H_{0} (Z)] = 1$ and $E [H_{m} (Z)] = 0$ for every $m \geq 1$ , by induction on $m$ : the case $m = 1$ is $E [Z] = 0$ from step 1.1, and for $m \geq 1$ the recurrence and step 2.1 give $E [H_{m + 1} (Z)] = E [Z H_{m} (Z)] - m E [H_{m - 1} (Z)] = E [H_{m}^{'} (Z)] - m E [H_{m - 1} (Z)] = m E [H_{m - 1} (Z)] - m E [H_{m - 1} (Z)] = 0$ , where step 1.2 identifies $H_{m}^{'} = m H_{m - 1}$ and the induction hypothesis handles $E [H_{m - 1}]$ .

4.1givenstep 1.2step 2.1step 3.1algebraF6

Orthogonality: put $a (M, n) : = E [H_{M} (Z) H_{n} (Z)]$ for $M, n \geq 0$ . We show $a (M, n) = n! δ_{M n}$ . First $a (0, n) = E [H_{n} (Z)] = δ_{0 n}$ by step 3.1. For $M = 1$ and $n \geq 1$ , step 2.1 gives $a (1, n) = E [Z H_{n}] = E [H_{n}'] = n a (0, n - 1)$ . For $M \geq 2$ and $n \geq 1$ , the recurrence $H_{M} = x H_{M - 1} - (M - 1) H_{M - 2}$ , the Stein identity of step 2.1 applied to $p = H_{M - 1} H_{n}$ and the derivative relation of step 1.2 give $a (M, n) = E [Z H_{M - 1} H_{n}] - (M - 1) a (M - 2, n) = E [(H_{M - 1} H_{n})^{'}] - (M - 1) a (M - 2, n) = (M - 1) a (M - 2, n) + n a (M - 1, n - 1) - (M - 1) a (M - 2, n) = n a (M - 1, n - 1)$ , while for $n = 0$ and $M \geq 1$ one has $a (M, 0) = E [H_{M} (Z)] = 0$ by step 3.1. If $k \leq M$ and $k \leq n$ , iteration gives $a (M, n) = n (n - 1) \dots (n - k + 1) a (M - k, n - k)$ ; taking $k = n$ when $n \leq M$ yields $a (M, n) = n! a (M - n, 0)$ , which is $0$ for $M > n$ and is $n! a (0, 0) = n!$ for $M = n$ by step 3.1, while taking $k = M$ when $M < n$ yields $a (M, n) = n (n - 1) \dots (n - M + 1) a (0, n - M) = 0$ because $n - M \geq 1$ . Hence $a (M, n) = n! δ_{M n}$ , which is claim (i) together with step 3.1.

5.1givenstep 1.3step 2.2step 3.1step 4.1∎

Conclusion: step 3.1 and step 4.1 prove (i); step 1.3 proves (ii) (including the unitriangularity clause); step 2.2 proves (iii). No step used anything beyond the published derivative, moment and characteristic-function facts listed above.

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The Plancherel weights sum to one

Statement

For every $n \geq 0$ , $\sum_{λ ⊢ n} P_{n} (λ) = \frac{1}{n!} \sum_{λ ⊢ n} (f^{λ})^{2} = 1.$ Hence $P_{n}$ is a probability distribution on the finite set $Y_{n}$ (Finite probability spaces, outcome weights, events, and event probabilities): the weights are nonnegative and sum to one. In particular $\dim_{C} C [S_{n}] = n! = \sum_{λ ⊢ n} (f^{λ})^{2}$ is obtained in two independent ways.

Facts & Assumptions

Given: $n \geq 0$ ; the finite set $Y_{n}$ of partitions of $n$ and the weights $P_{n} (λ) = (f^{λ})^{2} / n!$ , where $f^{λ} = \dim_{C} S^{λ}$ is the number of standard $λ$ -tableaux and $P_{0} (\emptyset) = 1$ (The Plancherel measure on the partitions of $n$ ).

[F1]

If $G$ is a finite group and $k$ is algebraically closed with $char k ∤ ∣ G ∣$ , then there are finitely many irreducible representations $V_{1}, \dots, V_{r}$ of $G$ over $k$ , up to equivalence, and $k [G] ≅ V_{1}^{\oplus \dim V_{1}} \oplus \dots \oplus V_{r}^{\oplus \dim V_{r}}$ (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , there are finitely many irreducible representations, and each occurs in the regular representation with multiplicity equal to its degree).

[F2]

The character of $C [S_{n}]$ is $∣ S_{n} ∣$ at the identity and $0$ elsewhere, so $\dim_{C} C [S_{n}] = n!$ (The regular character is $∣ G ∣$ at $1$ and $0$ away from $1$ ).

[F3]

For $G = S_{n}$ over $C$ , the irreducible representations up to equivalence are exactly the Specht modules $S^{λ}$ , $λ ⊢ n$ (Specht modules classify the complex irreducibles of $S_{n}$ , Complex Specht modules are irreducible), and $\dim_{C} S^{λ} = f^{λ}$ (Standard polytabloids form a basis of a complex Specht module).

[F4]

$\sum_{λ ⊢ n} (f^{λ})^{2} = n!$ for every $n \geq 0$ , by the Robinson-Schensted count (The sum of squares of the standard tableau numbers, The Robinson-Schensted correspondence).

[F5]

A finite probability space is a finite set $Ω$ with weights $w \geq 0$ satisfying $\sum_{ω} w (ω) = 1$ (Finite probability spaces, outcome weights, events, and event probabilities).

Proof

technique · direct

1.1givenF1F2F3algebra

Regular-representation count: $C$ is algebraically closed of characteristic $0$ , and $∣ S_{n} ∣ = n! > 0$ , so $char C = 0$ does not divide $∣ S_{n} ∣$ , so [F1] applies to $G = S_{n}$ , $k = C$ : the regular representation is $C [S_{n}] ≅ ⨁_{i = 1}^{r} V_{i}^{\oplus \dim V_{i}}$ with $V_{1}, \dots, V_{r}$ representing the irreducible complex representations of $S_{n}$ up to equivalence. By [F3] this list is ${S^{λ} : λ ⊢ n}$ and $\dim_{C} S^{λ} = f^{λ}$ , so $C [S_{n}] ≅ ⨁_{λ ⊢ n} (S^{λ})^{\oplus f^{λ}}$ . Taking dimensions, which are additive over direct sums and multiplicative over direct powers, and using [F2] gives $n! = \dim_{C} C [S_{n}] = \sum_{λ ⊢ n} f^{λ} \cdot \dim_{C} S^{λ} = \sum_{λ ⊢ n} (f^{λ})^{2}$ .

1.2givenF4

Independent count: the same identity $\sum_{λ ⊢ n} (f^{λ})^{2} = n!$ is proved independently from the Robinson-Schensted bijection by [F4], so the two computations of $\dim_{C} C [S_{n}]$ agree without either appealing to the other.

2.1givenF5step 1.1step 1.2algebra

Normalization: dividing the identity of steps 1.1 and 1.2 by the positive integer $n!$ (for $n = 0$ both sides read $1 = 1$ and $P_{0} (\emptyset) = 1$ ) gives $\sum_{λ ⊢ n} P_{n} (λ) = 1$ . Each $P_{n} (λ) = (f^{λ})^{2} / n!$ is a quotient of a nonnegative integer by a positive integer, hence is $\geq 0$ , and $Y_{n}$ is finite (The Plancherel measure on the partitions of $n$ ); therefore $P_{n}$ , viewed as a function on the finite set $Y_{n}$ , satisfies both requirements of a finite probability space in [F5].

3.1givenstep 2.1algebra∎

Conclusion: the displayed normalization, the nonnegativity of the weights and the finite nonempty outcome set $Y_{n}$ are exactly the assertion that $P_{n}$ is a probability distribution on $Y_{n}$ ; the two independent evaluations computing $\dim_{C} C [S_{n}] = n!$ are steps 1.1 and 1.2. This holds for every $n \geq 0$ , including the degenerate case $n = 0$ with the single empty partition.

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The multivariate method of moments for a determinate limit

Statement

Assume AC. Let $d \geq 1$ , let $X_{n}$ be $R^{d}$ -valued random vectors with laws $μ_{n}$ , let $μ$ be a Borel probability on $R^{d}$ with all mixed moments finite, and suppose:

(i) for every multi-index $α$ , $E [\prod_{i} x_{i}^{α_{i}}]$ evaluated at $X_{n}$ converges to $\int \prod_{i} x_{i}^{α_{i}} d μ$ ;

(ii) $μ$ is determined among Borel probabilities with finite moments by its mixed moments.

Then $μ_{n} \Rightarrow μ$ weakly. In particular every multivariate Gaussian law $N_{d} (m, Σ)$ is moment-determinate: a Borel probability with the same mixed moments as $N_{d} (m, Σ)$ equals it.

Facts & Assumptions

Given: AC; a positive integer $d$ ; $R^{d}$ -valued random vectors $X_{n}$ , $n \geq 1$ , with laws $μ_{n}$ ; a Borel probability $μ$ on $R^{d}$ with $\int \prod_{i} ∣ x_{i} ∣^{α_{i}} d μ < \infty$ for every multi-index $α \in N^{d}$ , which satisfies (i) and (ii) of the Statement. For a multi-index $α$ write $∣ α ∣ = α_{1} + \dots + α_{d}$ and $w^{α} = \prod_{i} w_{i}^{α_{i}}$ .

[F1]

$μ_{n} \Rightarrow μ$ means $\int f d μ_{n} \to \int f d μ$ for every bounded continuous real $f$ ; a family is tight when one compact set captures mass $1 - ε$ from every member; a tight sequence of Borel probabilities on a Polish space has a weakly convergent subsequence, and a Polish space is a complete separable metric space (Weak convergence of borel probability measures, Tight family of probability measures, Prokhorov tightness theorem on polish spaces, Tightness extracts a weakly convergent subsequence).

[F2]

If $Y \geq 0$ and $a > 0$ then $P (Y \geq a) \leq E [Y] / a$ (Markov's inequality for random variables).

[F3]

If $μ_{k} \Rightarrow μ$ on a Polish $S$ , there are random elements on a common probability space with laws $μ_{k}, μ$ and converging almost surely (Skorokhod representation on polish spaces).

[F4]

On a finite measure space, almost sure convergence implies convergence in measure (On a finite measure space, almost-everywhere convergence implies convergence in measure), and if $f_{k} \to f$ in measure with ${f_{k}}$ uniformly integrable then $f_{k} \to f$ in $L^{1}$ , so $f$ is integrable and the expectations converge (Vitali convergence theorem on finite and sigma-finite measure spaces, A uniformly integrable family); moreover a family bounded in $L^{2}$ is uniformly integrable according to that definition, because $\int_{{∣ f ∣ > M}} ∣ f ∣ \leq M^{- 1} \sup_{k} E f_{k}^{2}$ .

[F5]

If $X, Y \in L^{2}$ then $E ∣ X Y ∣ \leq (E X^{2})^{1 / 2} (E Y^{2})^{1 / 2}$ (Cauchy-Schwarz for random variables).

[F6]

For a multivariate normal $Y \sim N_{d} (m, Σ)$ and $u \in R^{d}$ , the projection $u \cdot Y$ is normal with mean $u \cdot m$ and variance $u^{T} Σ u$ , its characteristic function is $\exp (i t u \cdot m - \frac{1}{2} t^{2} u^{T} Σ u)$ , and two Borel probabilities on $R^{d}$ whose one-dimensional projections all have the same laws are equal (Multivariate normal law, including singular covariance, Characteristic function of a multivariate normal law, Cramer wold device).

[F7]

If $Z \sim N (0, 1)$ then $E [Z^{2 k}] = (2 k - 1)!! < \infty$ for $k \geq 1$ , so $E ∣ Z ∣^{k} < \infty$ for every $k$ (Gaussian even moments for Brownian increments; for odd $k$ use $∣ Z ∣^{k} \leq 1 + Z^{k + 1}$ and the even bound at $k + 1$ ).

[F8]

If $Y$ has all moments and $E [Y^{k}] = E [Z^{k}]$ for all $k$ with $Z \sim N (0, 1)$ , then $Y \sim N (0, 1)$ (The standard Gaussian law is determined by its moments).

[F9]

Multinomial expansion: $(\sum_{i = 1}^{d} u_{i})^{⁣ k} = \sum_{∣ α ∣ = k} (\binom{k}{α}) u^{α}$ , with $(\binom{k}{α}) = \frac{k!}{α!}$ , for all real $u_{i}$ (The multinomial coefficient equals $n! / \prod_{i < m} k_{i}!$ , and $(x_{0} + \dots + x_{m - 1})^{n} = \sum ι ⁣ (\binom{n}{k}) \prod_{i < m} x_{i}^{k_{i}}$ in $R$ ).

[F10]

Euclidean $R^{d}$ is complete ( $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ ) and separable: the rational coordinates have an enumeration ( $Q$ is countably infinite), and their $d$ -tuples can be enumerated by listing for each integer $M$ the finitely many tuples of enumeration indices at most $M$ . These vectors are dense: choose each rational coordinate within $ε / \sqrt{d}$ of the given coordinate using The rationals embed densely in the reals, giving Euclidean distance less than $ε$ . Hence it is Polish (Polish spaces are separable completely metrizable spaces). Closed cubes are compact by Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, and finite probability union bounds are supplied by Basic identities for a probability measure.

[F11]

Expectations of integrable variables are linear, monotone for real variables, and satisfy $∣ E U ∣ \leq E ∣ U ∣$ (Linearity, monotonicity, and the modulus bound for expectation).

Proof

technique · direct

1.1givenF6F7F9algebraF11

Gaussian projections: let $Y \sim N_{d} (m, Σ)$ , $u \in R^{d}$ , $c : = u \cdot m$ and $s : = u^{T} Σ u \geq 0$ . By [F6] the projection $u \cdot Y$ is normal with mean $c$ and variance $s$ , and by [F7] the standard normal has moments of every order; hence $E ∣ u \cdot Y ∣^{k} < \infty$ for every $k$ , and by [F9] and linearity of expectation, $E [(u \cdot Y)^{k}] = \sum_{∣ α ∣ = k} (\binom{k}{α}) u^{α} E [Y^{α}]$ , a finite sum of finite mixed moments. Indeed each coordinate has all absolute moments by [F7]; for a multi-index of total degree $q > 0$ , $\prod_{i} ∣ Y_{i} ∣^{α_{i}} \leq \max_{i} ∣ Y_{i} ∣^{q} \leq \sum_{i} ∣ Y_{i} ∣^{q}$ , proving mixed absolute integrability. The degree-zero product is $1$ .

1.2givenF2F10algebra

Tightness of the sequence: for each coordinate $i$ , hypothesis (i) applied to the multi-index with a single $2$ in place of $i$ gives $E [X_{n, i}^{2}] \to \int x_{i}^{2} d μ$ , so $C : = \max_{i} \sup_{n} E [X_{n, i}^{2}] < \infty$ . Fix $ε > 0$ and choose $R > 0$ with $d C / R^{2} < ε$ ; the cube $K : = [- R, R]^{d}$ is compact by [F10] and $R^{d} ∖ K$ is contained in the union of the $d$ coordinate slabs ${∣ x_{i} ∣ > R}$ , so by [F2] and the union bound of [F10], $μ_{n} (R^{d} ∖ K) \leq \sum_{i = 1}^{d} μ_{n} ({∣ x_{i} ∣ > R}) \leq \sum_{i = 1}^{d} E [X_{n, i}^{2}] / R^{2} \leq d C / R^{2} < ε$ for every $n$ . Hence ${μ_{n}}$ is tight.

2.1givenF5F9step 1.1algebraF11

Matching of projection moments: let $ν$ be a Borel probability on $R^{d}$ with the same mixed moments as $Y$ , i.e. $\int w^{α} d ν = E [Y^{α}]$ for every multi-index $α$ , and let $W \sim ν$ . Then for every $u \in R^{d}$ and every $k \geq 0$ the multinomial expansion [F9] gives $E [(u \cdot W)^{k}] = \sum_{∣ α ∣ = k} (\binom{k}{α}) u^{α} \int w^{α} d ν = \sum_{∣ α ∣ = k} (\binom{k}{α}) u^{α} E [Y^{α}] = E [(u \cdot Y)^{k}]$ by step 1.1; moreover $E [(u \cdot W)^{2 k}] = E [(u \cdot Y)^{2 k}] < \infty$ , so [F5] gives $E ∣ u \cdot W ∣^{k} \leq (E [(u \cdot W)^{2 k}])^{1 / 2} < \infty$ . Thus all moments of the projection $u \cdot W$ are finite and equal those of $u \cdot Y$ .

2.2givenF1F3F10step 1.2

Extraction along any subsequence: let $(μ_{n_{k}})$ be an arbitrary subsequence of $(μ_{n})$ . By step 1.2 the subfamily ${μ_{n_{k}}}$ is tight as well, so by [F1], applicable to the Polish space verified in [F10], it has a further subsequence $μ_{n_{k_{j}}}$ converging weakly to some Borel probability $μ^{''}$ on $R^{d}$ ; by [F3] there are random elements $Y_{j}, Y$ on a common probability space with laws $μ_{n_{k_{j}}}, μ^{''}$ and $Y_{j} \to Y$ almost surely.

3.1givenF6F8step 1.1step 2.1algebraF11F2F10

Projections determine the Gaussian: keep the notation of steps 1.1 and 2.1 with $W \sim ν$ and fix $u \in R^{d}$ . If $s = 0$ then, by step 2.1 with $k = 1, 2$ , $E [u \cdot W] = c$ and $E [(u \cdot W)^{2}] = E [(u \cdot Y)^{2}] = s + c^{2} = c^{2}$ , so $Var (u \cdot W) = E [(u \cdot W - c)^{2}] = 0$ and $u \cdot W = c$ almost surely: by [F2], $P (∣ u \cdot W - c ∣ > 1 / l) = 0$ for every integer $l \geq 1$ , and their countable union is the event $u \cdot W \neq c$ , of probability zero by [F10]. This is the law of $u \cdot Y$ . If $s > 0$ , put $Z^{'} : = (u \cdot W - c) / \sqrt{s}$ ; by step 2.1 its moments satisfy $E [(Z^{'})^{k}] = s^{- k / 2} \sum_{j = 0}^{k} (\binom{k}{j}) (- c)^{k - j} E [(u \cdot W)^{j}] = s^{- k / 2} \sum_{j = 0}^{k} (\binom{k}{j}) (- c)^{k - j} E [(u \cdot Y)^{j}] = E [ζ^{k}]$ for $ζ \sim N (0, 1)$ , because $(u \cdot Y - c) / \sqrt{s} \sim N (0, 1)$ by [F6]; and $E ∣ Z^{'} ∣^{k} < \infty$ by step 2.1. [F8] therefore gives $Z^{'} \sim N (0, 1)$ , so $u \cdot W \sim N (c, s)$ , again the law of $u \cdot Y$ .

3.2givenF4step 2.2algebra

Uniform integrability along the coupling: fix a multi-index $α$ and let $f_{j} : = \prod_{i} Y_{j, i}^{α_{i}}$ , so $f_{j} \to \prod_{i} Y_{i}^{α_{i}}$ almost surely by step 2.2. By hypothesis (i) applied to the multi-index $2 α$ , $E [f_{j}^{2}] = E [\prod_{i} X_{n_{k_{j}}, i}^{2 α_{i}}] \to \int \prod_{i} x_{i}^{2 α_{i}} d μ$ , so $\sup_{j} E f_{j}^{2} < \infty$ ; hence $\int_{{∣ f_{j} ∣ > M}} ∣ f_{j} ∣ \leq M^{- 1} \sup_{j} E f_{j}^{2}$ tends to $0$ uniformly in $j$ as $M \to \infty$ , and the family ${f_{j}}$ is uniformly integrable by [F4].

4.1givenF6step 3.1

Gaussian determinacy: if $ν$ has the same mixed moments as $Y \sim N_{d} (m, Σ)$ , then by step 3.1 every projection $u \cdot W$ of $W \sim ν$ has the law of the projection $u \cdot Y$ ; by the Cramér-Wold clause of [F6] the laws of $W$ and $Y$ coincide, so $ν = N_{d} (m, Σ)$ .

4.2givenstep 2.2step 3.2algebra

Identification of the limit: with the notation of step 3.2, almost sure convergence implies convergence in measure on the finite measure space by [F4]; together with the uniform integrability of step 3.2, Vitali's theorem [F4] gives $E [\prod_{i} Y_{j, i}^{α_{i}}] \to E [\prod_{i} Y_{i}^{α_{i}}]$ and shows the limit is finite. The left-hand side equals $E [\prod_{i} X_{n_{k_{j}}, i}^{α_{i}}]$ , which tends to $\int \prod_{i} x_{i}^{α_{i}} d μ$ by hypothesis (i); hence $\int w^{α} d μ^{''} = \int w^{α} d μ$ for every multi-index $α$ , $μ^{''}$ has finite mixed moments, and hypothesis (ii) gives $μ^{''} = μ$ .

5.1givenF1step 4.2algebra

Convergence of the full sequence: let $(μ_{n_{k}})$ be an arbitrary subsequence of $(μ_{n})$ . Steps 2.2, 3.2 and 4.2 applied to it produce a further subsequence converging weakly to $μ$ . Hence $μ_{n} \Rightarrow μ$ : otherwise there are a bounded continuous real function $f$ on $R^{d}$ , a real $ε > 0$ and a subsequence with $∣ \int f d μ_{n_{k}} - \int f d μ ∣ \geq ε$ for all $k$ (Weak convergence of borel probability measures), yet that subsequence has a further subsequence converging weakly to $μ$ , along which $\int f d μ_{n_{k_{j}}} \to \int f d μ$ , a contradiction.

6.1givenstep 4.1step 5.1∎

Conclusion: step 5.1 proves the convergence assertion from hypotheses (i) and (ii), and steps 1.1, 2.1, 3.1 and 4.1 prove that every multivariate Gaussian law is moment-determinate. AC was used exactly through the Polish-space existence theorems of [F1], [F3] and the determinacy lemma [F8].

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The RSK shape of a uniform random permutation has the Plancherel law

Statement

Let $n \geq 1$ and let $σ$ be uniformly distributed on $S_{n}$ (The uniform probability space on a nonempty finite set). Let $sh (σ) ⊢ n$ be the common shape of the Robinson-Schensted pair $(P (σ), Q (σ))$ (The Robinson-Schensted correspondence). Then $sh$ is a random element with values in the finite measurable space $(Y_{n}, 2^{Y_{n}})$ (Law or distribution of a random element) and for every $λ ⊢ n$ $P (sh (σ) = λ) = \frac{(f^{λ})^{2}}{n!} = P_{n} (λ) .$ In particular the uniform distribution on $S_{n}$ pushes forward to the Plancherel measure of order $n$ , and the length of a longest increasing subsequence of $σ$ has the same law as the first row length of a Plancherel-random diagram.

Facts & Assumptions

Given: $n \geq 1$ ; the permutations of ${1, \dots, n}$ written in one-line form, equipped with the uniform probability; the Robinson-Schensted map $σ \mapsto (P (σ), Q (σ))$ ; the shape $sh (σ)$ ; the number $f^{λ}$ of standard $λ$ -tableaux for $λ ⊢ n$ ; and the Plancherel weights $P_{n} (λ) = (f^{λ})^{2} / n!$ (The Plancherel measure on the partitions of $n$ ).

[F1]

The Robinson-Schensted map is a bijection from the permutations of ${1, \dots, n}$ onto the set of pairs $(P, Q)$ of standard tableaux of the same shape $λ ⊢ n$ ; $P (σ)$ has shape $sh (σ)$ (The Robinson-Schensted correspondence).

[F2]

For every $λ ⊢ n$ the number of standard $λ$ -tableaux equals $f^{λ}$ , the number of paths from the empty diagram to $λ$ in the Young graph (Young-graph paths correspond to standard tableaux, Standard polytabloids form a basis of a complex Specht module).

[F3]

On a nonempty finite set the uniform probability space gives every element weight $1 / ∣ Ω ∣$ , so an event of cardinality $m$ has probability $m / ∣ Ω ∣$ (The uniform probability space on a nonempty finite set, Finite probability spaces, outcome weights, events, and event probabilities).

[F4]

A function from a finite probability space to a finite set is a random element, its law being the pushforward of the probability (Law or distribution of a random element); a real-valued such function is a real random variable with the distribution of Real random variables on finite probability spaces and their finite distributions.

[F5]

If $σ$ has insertion tableau $P (σ)$ of shape $λ$ , then the length of a longest increasing subsequence of $σ$ is $λ_{1}$ and the length of a longest decreasing subsequence is $λ_{1}^{'}$ (The Schensted theorem on longest increasing and decreasing subsequences).

Proof

technique · direct

1.1givenF1F2

Fibres of the shape map: by [F1] the Robinson-Schensted map is a bijection from the set of $n!$ permutations of ${1, \dots, n}$ onto the set of pairs $(P, Q)$ of standard tableaux of equal shape $λ ⊢ n$ . For a fixed $λ ⊢ n$ the permutations with $sh (σ) = λ$ correspond bijectively to the pairs $(P, Q)$ of standard $λ$ -tableaux, and by [F2] there are exactly $f^{λ}$ choices for $P$ and independently $f^{λ}$ choices for $Q$ ; hence the fibre over $λ$ has cardinality $∣ {sh = λ} ∣ = (f^{λ})^{2}$ .

2.1givenF3step 1.1algebra

Probability of a shape: on $S_{n}$ the uniform probability space of [F3] assigns weight $1 / n!$ to every permutation, so the event ${sh = λ}$ of step 1.1 has probability $∣ {sh = λ} ∣ / n! = (f^{λ})^{2} / n! = P_{n} (λ)$ ; the denominator $n!$ is positive for $n \geq 1$ .

3.1givenF4step 2.1

Random element and its law: $sh$ is a function from the finite probability space $S_{n}$ to the finite set $Y_{n}$ , hence by [F4] a random element with values in $Y_{n}$ , and its law is the pushforward of the uniform probability; step 2.1 computes that law to be exactly $P_{n}$ . Every subset of $Y_{n}$ has a measurable inverse image, since every subset of the finite outcome space $S_{n}$ is an event. Thus the partition-valued map itself has the law $P_{n}$ ; a real encoding would instead have the corresponding encoded law.

4.1givenF5step 2.1step 3.1algebra∎

Longest increasing subsequence: for every realisation $σ$ , [F5] identifies the length of a longest increasing subsequence of $σ$ with the first row length $sh (σ)_{1}$ . Therefore, for every $l \geq 0$ , the probability that the longest increasing subsequence has length $l$ equals $P (sh (σ)_{1} = l) = P_{n} ({λ : λ_{1} = l})$ by step 3.1, which is precisely the law of the first row length of a diagram drawn from $P_{n}$ . Together with step 3.1 this proves the statement.

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The Logan-Shepp-Vershik-Kerov limit profile $Ω$

Definition

Define the function $Ω : R \to R$ by $Ω (x) = \frac{2}{π} (x \arcsin \frac{x}{2} + \sqrt{4 - x^{2}}) (∣ x ∣ \leq 2), Ω (x) = ∣ x ∣ (∣ x ∣ \geq 2),$ with the principal arcsine of Principal inverse sine and inverse cosine. Its elementary properties, all used below, are as follows.

(a) Evenness. The functions $x \mapsto x \arcsin (x / 2)$ , $x \mapsto \sqrt{4 - x^{2}}$ and $x \mapsto ∣ x ∣$ are even, so $Ω$ is even.

(b) Values and continuity at the junctions. At $x = \pm 2$ the first formula gives $\frac{2}{π} (\pm 2 \arcsin (\pm 1) + 0) = 2$ because $\arcsin (\pm 1) = \pm π / 2$ , agreeing with $∣ x ∣ = 2$ ; the arcsine branch and $∣ x ∣$ are continuous on their closed domains, and the two branches agree at the two junction points, so $Ω$ is continuous on all of $R$ .

(c) First derivative. For $∣ x ∣ < 2$ differentiability of $\arcsin$ on $(- 1, 1)$ (Principal inverse sine and inverse cosine, Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f^{'} (c) \neq 0$ , then the inverse $g$ is differentiable at $f (c)$ with $g^{'} (f (c)) = 1 / f^{'} (c)$ ; and if $f^{'} (c) = 0$ then $g$ is not differentiable at $f (c)$ , The derivatives of sine and cosine are cosine and minus sine) and the chain and product rules (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ) applied to $x \mapsto \frac{2}{π} (x \arcsin \frac{x}{2} + \sqrt{4 - x^{2}})$ give $Ω^{'} (x) = \frac{2}{π} (\arcsin \frac{x}{2} + \frac{x}{2} (1 - \frac{x^{2}}{4})^{- 1 / 2} - \frac{x}{2} (1 - \frac{x^{2}}{4})^{- 1 / 2}) = \frac{2}{π} \arcsin \frac{x}{2} .$ For $∣ x ∣ > 2$ the derivative of the restriction is $Ω^{'} (x) = sgn x$ . As $x \to 2^{-}$ the formula tends to $\frac{2}{π} \cdot \frac{π}{2} = 1 = sgn x$ for $x > 2$ , and as $x \to - 2^{+}$ it tends to $- 1 = sgn x$ for $x < - 2$ ; hence $Ω$ is differentiable at every real point with $Ω^{'} (x) = \frac{2}{π} \arcsin \frac{x}{2} (∣ x ∣ < 2), Ω^{'} (x) = sgn x (∣ x ∣ > 2),$ and the one-sided derivatives at $x = \pm 2$ both equal $\pm 1$ (they are the limits of $Ω^{'}$ from within $(- 2, 2)$ and from outside).

(d) Lipschitz bound and smoothness. Since $∣ \arcsin y ∣ < π / 2$ for $∣ y ∣ < 1$ , one has $∣ Ω^{'} (x) ∣ < 1$ for $∣ x ∣ < 2$ , while $∣ Ω^{'} (x) ∣ = 1$ for $∣ x ∣ > 2$ . On each of the intervals $(- \infty, - 2]$ , $[- 2, 2]$ , $[2, \infty)$ the function is continuous and differentiable on the interior, so the mean value theorem (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ) gives $∣ Ω (x) - Ω (y) ∣ \leq ∣ x - y ∣$ for $x, y$ in the same interval, and the continuity at $\pm 2$ gives the same bound across the junctions; thus $Ω$ is $1$ -Lipschitz. On $(- 2, 2)$ the arcsine branch is $C^{\infty}$ with $Ω^{''} (x) = \frac{2}{π} (4 - x^{2})^{- 1 / 2} > 0,$ so $Ω$ is $C^{\infty}$ there with $Ω^{'}$ strictly increasing. Since $∣ Ω (x) - Ω (y) ∣ \leq ∣ x - y ∣$ for all $x, y$ and $Ω (x) = ∣ x ∣$ for $∣ x ∣ \geq 2$ , we have $Ω \in D_{0}$ in the sense of Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram, with $σ_{Ω} = \frac{1}{2} (Ω - ∣ x ∣)$ supported in $[- 2, 2]$ and $σ_{Ω} (0) = 2 / π > 0$ . No choice principle is used.

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Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$

Definition

(a) Shifted character observables. For a partition $ρ ⊢ r$ (Partitions, English diagrams, and conjugation) and $λ ⊢ n$ set $p_{ρ}^{#} (λ) : = {\begin{cases} n^{↓ r} \frac{χ_{ρ \cup 1^{n - r}}^{λ}}{\dim_{C} S^{λ}}, & n \geq r, \\ 0, & n < r, \end{cases}$ where $n^{↓ r} = n (n - 1) \dots (n - r + 1)$ is the falling factorial of The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , $ρ \cup 1^{n - r} = (ρ, 1, \dots, 1) ⊢ n$ is the padded partition, $χ^{λ}$ is the complex irreducible character of $S_{n}$ indexed by $λ$ , and $\dim_{C} S^{λ} = χ_{(1^{n})}^{λ} = f^{λ} > 0$ by Standard polytabloids form a basis of a complex Specht module. Character values are the power-sum coefficients, $χ^{λ} (μ) = ⟨ s_{λ}, p_{μ} ⟩$ for $μ ⊢ n$ (Irreducible symmetric-group character values are power-sum coefficients). For $n \geq r$ the quotient is a finite real number: a permutation and its inverse have the same cycle type and are conjugate (reverse the order within each cycle), while a complex character satisfies $χ (g^{- 1}) = \overline{χ (g)}$ and is constant on conjugacy classes (For a complex character, $χ (1) = \dim V$ , $χ$ is a class function, and $∣ χ (g) ∣ \leq χ (1)$ with equality exactly at scalars). Therefore $χ (g) = \overline{χ (g)}$ . The definition for $n < r$ is consistent with $n^{↓ r} = 0$ : in particular $p_{1}^{#} (λ) = n^{↓ 1} \frac{χ_{(1^{n})}^{λ}}{\dim_{C} S^{λ}} = n \frac{f^{λ}}{f^{λ}} = n (λ ⊢ n, n \geq 1) .$

(b) Profile moments. For $ω \in D_{0}$ with profile $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ (Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram) and $k \geq 2$ define the profile moment ${\tilde{p}}_{k} [ω] : = k (k - 1) \int_{R} x^{k - 2} σ_{ω} (x) d x, {\tilde{p}}_{1} [ω] : = 0.$ The integrand is continuous, since $σ_{ω}$ is Lipschitz, and compactly supported. Its integral is the Riemann integral over any compact interval containing its support, so it exists and is finite by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. If in addition $ω$ is piecewise linear with finitely many corners, $σ_{ω}$ is continuous, compactly supported and piecewise $C^{1}$ , and applying integration by parts on each linear piece and summing (the boundary terms cancel, since $σ_{ω}$ vanishes at the ends and its values at the interior corners enter twice with opposite signs) gives ${\tilde{p}}_{k} [ω] = - k \int_{R} x^{k - 1} σ_{ω}^{'} (x) d x (k \geq 2),$ in agreement with the source's formula (2.2), where $σ_{ω}^{'}$ exists except at the finitely many corners; this is the form used for the Young-diagram profiles of this page. The convention ${\tilde{p}}_{1} = 0$ is the source's.

(c) Scaling. For every $s > 0$ and $k \geq 2$ , the substitution $y = s x$ (Monotone change of variable for Riemann-integrable functions, applied on a compact interval containing the support) in the definition of the $s$ -scaling $ω_{s} (x) = s^{- 1} ω (s x)$ gives ${\tilde{p}}_{k} [ω_{s}] = k (k - 1) \int_{R} x^{k - 2} \frac{1}{2} (s^{- 1} ω (s x) - ∣ x ∣) d x = k (k - 1) s^{- k} \int_{R} y^{k - 2} σ_{ω} (y) d y = s^{- k} {\tilde{p}}_{k} [ω],$ because $\frac{1}{2} (s^{- 1} ∣ s x ∣ - ∣ x ∣) = 0$ ; hence for $λ ⊢ n$ , $n \geq 1$ , the $\sqrt{n}$ -scaled profile $\overset{ˉ}{λ}$ of Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram satisfies ${\tilde{p}}_{k} [\overset{ˉ}{λ}] = n^{- k / 2} {\tilde{p}}_{k} [λ (\cdot)] .$ No choice principle is used.

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The RSK union bound localizes Plancherel profiles

Statement

Let $n \geq 1$ , let $σ$ be uniform on $S_{n}$ and let $λ = sh (σ) ⊢ n$ . For every integer $L$ with $1 \leq L \leq n$ , $P (λ_{1} \geq L) \leq \frac{(\binom{n}{L})}{L!} \leq (\frac{e^{2} n}{L^{2}})^{L},$ and the same two inequalities hold for $λ_{1}^{'}$ . Consequently, for every constant $C > e$ there is $n_{0}$ such that for all $n \geq n_{0}$ $P (λ_{1} \leq C \sqrt{n} and λ_{1}^{'} \leq C \sqrt{n}) \geq 1 - 2 (\frac{e^{2}}{C^{2}})^{⌊ C \sqrt{n} ⌋ - 1} ⟶ 1,$ and on the event in question the function $x \mapsto \overset{ˉ}{λ} (x) - ∣ x ∣$ is supported in the fixed compact interval $[- C, C]$ .

Facts & Assumptions

Given: $n \geq 1$ ; $σ$ uniformly distributed on the permutations of ${1, \dots, n}$ ; $λ = sh (σ)$ the Robinson-Schensted shape, a random variable with law $P_{n}$ (The RSK shape of a uniform random permutation has the Plancherel law); an integer $L$ with $1 \leq L \leq n$ .

[F1]

The length of a longest increasing subsequence of $σ$ is $λ_{1}$ and the length of a longest decreasing subsequence is $λ_{1}^{'}$ (The Schensted theorem on longest increasing and decreasing subsequences); the uniform probability on $S_{n}$ gives every permutation weight $1 / n!$ (The uniform probability space on a nonempty finite set, Finite probability spaces, outcome weights, events, and event probabilities).

[F2]

Probability is subadditive: $P (⋃_{j} A_{j}) \leq \sum_{j} P (A_{j})$ for finitely many events (Basic identities for a probability measure).

[F3]

$(\binom{n}{L}) = \frac{n (n - 1) \dots (n - L + 1)}{L!} \leq \frac{n^{L}}{L!}$ for $0 \leq L \leq n$ ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ ); and for real $x \geq 0$ one has $e^{x} = \sum_{j \geq 0} x^{j} / j!$ , so $e^{L} \geq L^{L} / L!$ and hence $L! \geq (L / e)^{L}$ (The power-series, product-limit, IVP, functional-equation, and Picard definitions agree).

[F4]

For a partition $λ ⊢ n$ the support of $σ_{λ}$ is contained in $[- λ_{1}^{'}, λ_{1}]$ , and for the $\sqrt{n}$ -scaled profile $\overset{ˉ}{λ}$ one has $σ_{\overset{ˉ}{λ}} (x) = n^{- 1 / 2} σ_{λ} (\sqrt{n} x)$ (Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram).

Proof

technique · direct

1.1givenF1F2algebra

Union bound: by [F1] the event ${λ_{1} \geq L}$ is contained in the union, over the $(\binom{n}{L})$ subsets $I \subseteq {1, \dots, n}$ of cardinality $L$ , of the event $A_{I}$ that the values $(σ_{i})_{i \in I}$ are increasing in the order of $I$ . For a fixed $I$ , the relative order of the $L$ distinct values $(σ_{i})_{i \in I}$ is uniform over the $L!$ orders, by symmetry of the uniform permutation (each ordering of the values on $I$ is realised by exactly $n! / L!$ permutations); hence $P (A_{I}) = 1 / L!$ , and [F2] gives $P (λ_{1} \geq L) \leq (\binom{n}{L}) / L!$ . Replacing $σ$ by the reversed word, whose uniform law is again uniform on $S_{n}$ and whose longest increasing subsequences are exactly the reversed longest decreasing subsequences of $σ$ , the same computation with [F1] gives $P (λ_{1}^{'} \geq L) \leq (\binom{n}{L}) / L!$ .

1.2givenF4algebra

Support: by [F4] the support of $σ_{λ}$ lies in $[- λ_{1}^{'}, λ_{1}]$ , and $σ_{\overset{ˉ}{λ}} (x) = n^{- 1 / 2} σ_{λ} (\sqrt{n} x)$ ; hence if $λ_{1} \leq C \sqrt{n}$ and $λ_{1}^{'} \leq C \sqrt{n}$ , then $σ_{\overset{ˉ}{λ}}$ is supported in $[- C, C]$ , and so is $x \mapsto \overset{ˉ}{λ} (x) - ∣ x ∣ = 2 σ_{\overset{ˉ}{λ}} (x)$ .

2.1givenF3step 1.1algebra

Arithmetic bound: by [F3], $(\binom{n}{L}) / L! \leq n^{L} / (L!)^{2} \leq n^{L} / (L / e)^{2 L} = (e^{2} n / L^{2})^{L}$ ; combined with step 1.1 this proves both displayed inequalities.

3.1givenF2step 1.1step 2.1algebra

Localization: let $C > e$ and put $L_{n} : = ⌊ C \sqrt{n} ⌋ + 1$ , so $L_{n} > C \sqrt{n} \geq 1$ ; for all sufficiently large $n$ one has $L_{n} \leq n$ . Since ${λ_{1} > C \sqrt{n}} \subseteq {λ_{1} \geq L_{n}}$ , steps 1.1 and 2.1 give $P (λ_{1} > C \sqrt{n}) \leq (e^{2} n / L_{n}^{2})^{L_{n}} \leq (e^{2} / C^{2})^{L_{n}} \leq (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1}$ , where $e^{2} n / L_{n}^{2} \leq e^{2} / C^{2}$ uses $L_{n} > C \sqrt{n}$ and the last inequality uses $0 < e^{2} / C^{2} < 1$ and $L_{n} \geq ⌊ C \sqrt{n} ⌋ - 1$ ; the same bound holds with $λ_{1}^{'}$ in place of $λ_{1}$ . By [F2], $P (λ_{1} > C \sqrt{n} or λ_{1}^{'} > C \sqrt{n}) \leq 2 (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1}$ , so the probability of the complementary event ${λ_{1} \leq C \sqrt{n}$ and $λ_{1}^{'} \leq C \sqrt{n}}$ is at least $1 - 2 (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1}$ ; since $0 < e^{2} / C^{2} < 1$ , this lower bound tends to $1$ .

4.1givenstep 1.2step 3.1algebra∎

Conclusion: on the event ${λ_{1} \leq C \sqrt{n}$ and $λ_{1}^{'} \leq C \sqrt{n}}$ step 1.2 shows that $x \mapsto \overset{ˉ}{λ} (x) - ∣ x ∣$ is supported in the fixed compact interval $[- C, C]$ , and step 3.1 shows that this event has probability at least $1 - 2 (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1} \to 1$ .

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Joint convergence in distribution and the normalized cycle-character observables

Definition

Fix an integer $N \geq 2$ . For $n \geq 1$ and $2 \leq k \leq N$ define the real random variable $η_{k}^{(n)} (λ) : = \frac{p_{k}^{#} (λ)}{\sqrt{k} n^{k / 2}}, λ ⊢ n,$ on the finite probability space $(Y_{n}, P_{n})$ of The Plancherel measure on the partitions of $n$ , whose weights are nonnegative and sum to one by The Plancherel weights sum to one, where $p_{k}^{#}$ is the shifted character observable of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ; since $p_{k}^{#}$ is a real function on $Y_{n}$ , each $η_{k}^{(n)}$ is a real random variable on the finite space $Y_{n}$ . Let $η^{(n)} : = (η_{2}^{(n)}, \dots, η_{N}^{(n)}) : Y_{n} \to R^{N - 1}$ be the corresponding $R^{N - 1}$ -valued random element, with law the pushforward of $P_{n}$ (Law or distribution of a random element).

Joint convergence in distribution of such vectors means convergence in distribution of random elements in the sense of Convergence in distribution of random elements: if $η^{(n)}$ has law $μ_{n}$ and $η$ has law $μ$ on $R^{N - 1}$ , then $η^{(n)} ⟹ η$ means $μ_{n} \Rightarrow μ$ weakly, that is, $\int f d μ_{n} \to \int f d μ$ for every bounded continuous real function $f$ on $R^{N - 1}$ .

The standard Gaussian target law is $N_{N - 1} (0, I_{N - 1})$ (Multivariate normal law, including singular covariance), the law of a vector with independent standard normal coordinates; under AC this multi-dimensional law exists and is available in the library. For independent standard Gaussian random variables $ξ_{k}$ , $2 \leq k \leq N$ , this target is the law of $(ξ_{2}, \dots, ξ_{N})$ . The unnormalized observables $p_{k}^{#} / n^{k / 2}$ instead have the target coordinates $ζ_{k} : = \sqrt{k} ξ_{k}$ , independent centered Gaussians of variances $k$ ; the normalization $η_{k}^{(n)} = p_{k}^{#} / (\sqrt{k} n^{k / 2})$ is chosen so that this is the limit asserted in Kerov's central limit theorem for normalized cycle characters. No convergence is asserted in this definition, and no choice principle is used beyond the existence of the target law.

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The profile moments of $Ω$ are central binomial coefficients

Statement

For the limit profile $Ω$ of The Logan-Shepp-Vershik-Kerov limit profile $Ω$ , ${\tilde{p}}_{2 m} [Ω] = \frac{(2 m)!}{m! m!} = (\binom{2 m}{m}) (m \geq 1), {\tilde{p}}_{k} [Ω] = 0 (k odd),$ and ${\tilde{p}}_{1} [Ω] = 0$ by the declaration of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ .

Facts & Assumptions

Given: the even profile $Ω$ and its profile moments ${\tilde{p}}_{k} [ω] = k (k - 1) \int_{R} x^{k - 2} σ_{ω} (x) d x$ , $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ , Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ).

[F1]

$Ω$ is even, $Ω (x) = ∣ x ∣$ for $∣ x ∣ \geq 2$ , and for $∣ x ∣ < 2$ it is $C^{1}$ with $Ω^{'} (x) = \frac{2}{π} \arcsin \frac{x}{2}$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ). Hence $σ_{Ω} = \frac{1}{2} (Ω - ∣ x ∣)$ is even, vanishes outside $[- 2, 2]$ , and is continuous and piecewise $C^{1}$ on $[- 2, 0]$ and $[0, 2]$ , with $σ_{Ω} (\pm 2) = 0$ ; integrating by parts on the two pieces gives, for $k \geq 2$ , $\int_{R} x^{k - 1} σ_{Ω}^{'} (x) d x = - (k - 1) \int_{R} x^{k - 2} σ_{Ω} (x) d x$ (all boundary terms vanish: at $\pm 2$ because $σ_{Ω}$ vanishes there, at $0$ because $k - 1 \geq 1$ ), so $- k \int_{R} x^{k - 1} σ_{Ω}^{'} = k (k - 1) \int_{R} x^{k - 2} σ_{Ω} = {\tilde{p}}_{k} [Ω]$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ).

[F2]

Monotone change of variables: if $φ : [c, d] \to [a, b]$ is a monotone differentiable bijection with integrable derivative and $f$ is Riemann integrable on $[a, b]$ , then $\int_{a}^{b} f = \int_{c}^{d} (f \circ φ) ∣ φ^{'} ∣$ (Monotone change of variable for Riemann-integrable functions).

[F3]

Wallis integrals: $I_{2 m} = \int_{0}^{π / 2} \sin^{2 m} θ d θ = \frac{π}{2} \prod_{k = 1}^{m} \frac{2 k - 1}{2 k} = \frac{π}{2} \cdot \frac{(2 m - 1)!!}{(2 m)!!}$ for $m \geq 0$ (Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze); and $(2 m)! = 2^{m} m! (2 m - 1)!!$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F4]

Arcsine: $\arcsin (\sin θ) = θ$ for $θ \in [0, π / 2]$ (Principal inverse sine and inverse cosine), and $\sin$ and $\cos$ have the usual derivatives (The derivatives of sine and cosine are cosine and minus sine).

[F5]

Integration by parts on a closed interval: if $u, v$ are differentiable on $[a, b]$ with integrable derivatives then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

Proof

technique · direct

1.1givenF1algebra

Odd moments vanish: $σ_{Ω}$ is even and supported in $[- 2, 2]$ by [F1], so for odd $k \geq 3$ the integrand $x^{k - 2} σ_{Ω} (x)$ is odd and its integral over the symmetric interval vanishes; hence ${\tilde{p}}_{k} [Ω] = 0$ for odd $k$ by the definition, and ${\tilde{p}}_{1} [Ω] = 0$ by the convention.

1.2givenF1algebra

Reduction for even moments: fix $m \geq 1$ and put $k = 2 m$ . By [F1] and the integration-by-parts form of the profile moment, ${\tilde{p}}_{2 m} [Ω] = - 2 m \int_{R} x^{2 m - 1} σ_{Ω}^{'} (x) d x = - m \int_{- 2}^{2} x^{2 m - 1} (\frac{2}{π} \arcsin \frac{x}{2} - sgn x) d x .$ The integrand is even (odd factor times the odd function $\frac{2}{π} \arcsin (x / 2) - sgn x$ ), so the integral equals $2 m \int_{0}^{2} x^{2 m - 1} (1 - \frac{2}{π} \arcsin \frac{x}{2}) d x = \int_{0}^{2} (1 - \frac{2}{π} \arcsin \frac{x}{2}) d (x^{2 m})$ .

2.1givenF2F4step 1.2algebra

Substitution $x = 2 \sin θ$ : by [F2] applied to the increasing bijection $θ \mapsto 2 \sin θ$ from $[0, π / 2]$ onto $[0, 2]$ (with $2 \cos θ \geq 0$ and $\arcsin (\sin θ) = θ$ by [F4]), the integral of step 1.2 equals $\int_{0}^{π / 2} (1 - \frac{2}{π} θ) \cdot 4 m 2^{2 m - 1} \sin^{2 m - 1} θ \cos θ d θ .$

3.1givenF3step 2.1algebra

Integration by parts and Wallis: on $[0, π / 2]$ put $u (θ) : = 1 - \frac{2}{π} θ$ and $v (θ) : = \frac{\sin^{2 m} θ}{2 m}$ ; both are differentiable with continuous derivatives $u^{'} \equiv - \frac{2}{π}$ and $v^{'} = \sin^{2 m - 1} θ \cos θ$ , so [F5] gives $\int_{0}^{π / 2} u v^{'} = [u v]_{0}^{π / 2} - \int_{0}^{π / 2} u^{'} v = \frac{2}{π} \cdot \frac{1}{2 m} \int_{0}^{π / 2} \sin^{2 m} θ d θ = \frac{I_{2 m}}{π m},$ because $u (π / 2) = 0$ and $u (0) = 1$ while $v (0) = \sin 0 = 0$ . Multiplying by $4 m 2^{2 m - 1}$ and using [F3], ${\tilde{p}}_{2 m} [Ω] = \frac{2^{2 m + 1}}{π} I_{2 m} = \frac{2^{2 m + 1}}{π} \cdot \frac{π}{2} \cdot \frac{(2 m - 1)!!}{(2 m)!!} = 2^{2 m} \frac{(2 m - 1)!!}{2^{m} m!} = \frac{(2 m)!}{m! m!}$ , where the last equality is $(2 m)! = 2^{m} m! (2 m - 1)!!$ of [F3].

4.1givenstep 1.1step 3.1∎

Conclusion: step 1.1 gives the vanishing for odd $k$ (including ${\tilde{p}}_{1} = 0$ by convention) and steps 1.2, 2.1, 3.1 give ${\tilde{p}}_{2 m} [Ω] = (\binom{2 m}{m})$ for every $m \geq 1$ .

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Plancherel expectations of the shifted character observables

Statement

For every partition $ρ$ with $r = ∣ ρ ∣$ and every $n \geq 0$ , $E_{P_{n}} [p_{ρ}^{#}] = {\begin{cases} n^{↓ r}, & ρ = (1^{r}), \\ 0, & ρ \neq (1^{r}), \end{cases}$ where the case $n < r$ is included: then $p_{ρ}^{#} \equiv 0$ on $Y_{n}$ and $n^{↓ r} = 0$ . In particular $E_{P_{n}} [p_{ρ}^{#}] = O (n^{∣ ρ ∣})$ uniformly in $n$ , and if $m_{1} (ρ) = 0$ and $ρ \neq \emptyset$ then $E_{P_{n}} [p_{ρ}^{#}] = 0$ for every $n$ .

Facts & Assumptions

Given: a partition $ρ$ with $r = ∣ ρ ∣$ ; the shifted observables $p_{ρ}^{#} (λ) = n^{↓ r} χ_{ρ \cup 1^{n - r}}^{λ} / \dim_{C} S^{λ}$ for $λ ⊢ n$ , $n \geq r$ , and $p_{ρ}^{#} (λ) = 0$ for $n < r$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ); the Plancherel weights $P_{n} (λ) = (f^{λ})^{2} / n!$ with $f^{λ} = \dim_{C} S^{λ} = χ_{(1^{n})}^{λ}$ (The Plancherel measure on the partitions of $n$ ).

[F1]

For every $λ ⊢ n$ , $\dim_{C} S^{λ} = χ_{(1^{n})}^{λ} = f^{λ} > 0$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ).

[F2]

$C [S_{n}] ≅ ⨁_{λ ⊢ n} (S^{λ})^{\oplus f^{λ}}$ as $C [S_{n}]$ -modules, the irreducibles being the Specht modules up to equivalence (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , there are finitely many irreducible representations, and each occurs in the regular representation with multiplicity equal to its degree, Specht modules classify the complex irreducibles of $S_{n}$ ).

[F3]

The character of the regular representation is $χ_{r e g} (g) = n!$ for $g = e$ and $χ_{r e g} (g) = 0$ for $g \neq e$ (The regular character is $∣ G ∣$ at $1$ and $0$ away from $1$ ).

[F4]

The sum of the Plancherel weights is one; equivalently $\sum_{λ ⊢ n} (f^{λ})^{2} = n!$ (The Plancherel weights sum to one, The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

Proof

technique · direct

1.1givenF1F4algebra

Expectation by characters: for $n \geq r$ , expanding the expectation against the Plancherel weights and inserting the definition of $p_{ρ}^{#}$ gives $E_{P_{n}} [p_{ρ}^{#}] = \sum_{λ ⊢ n} \frac{n^{↓ r} χ_{ρ \cup 1^{n - r}}^{λ}}{f^{λ}} \cdot \frac{(f^{λ})^{2}}{n!} = \frac{n^{↓ r}}{n!} \sum_{λ ⊢ n} χ_{ρ \cup 1^{n - r}}^{λ} f^{λ};$ for $n < r$ both $p_{ρ}^{#}$ and $n^{↓ r}$ vanish, so the formula also gives $0$ there. All quantities are finite, and $n! > 0$ .

1.2givenF2F3algebra

The class sum: by [F2] the character of $C [S_{n}]$ is the class function $χ_{r e g} = \sum_{λ ⊢ n} f^{λ} χ^{λ}$ ; evaluated at a permutation of cycle type $μ ⊢ n$ and compared with [F3] this gives $\sum_{λ ⊢ n} f^{λ} χ_{μ}^{λ} = χ_{r e g} (μ) = {\begin{cases} n!, & μ = (1^{n}), \\ 0, & μ \neq (1^{n}), \end{cases}$ the class $(1^{n})$ being exactly the identity class.

2.1givenstep 1.1step 1.2algebra

Case evaluation: applying step 1.2 with $μ = ρ \cup 1^{n - r} ⊢ n$ in step 1.1, the sum is $n!$ precisely when $ρ \cup 1^{n - r} = (1^{n})$ , i.e. when $ρ = (1^{r})$ , and is $0$ otherwise; dividing by $n!$ and multiplying by $n^{↓ r}$ gives $E_{P_{n}} [p_{ρ}^{#}] = n^{↓ r}$ for $ρ = (1^{r})$ and $0$ otherwise, including $n < r$ by step 1.1.

3.1givenstep 2.1algebra∎

Consequences: $n^{↓ r} = n (n - 1) \dots (n - r + 1)$ is $0$ for $n < r$ and of modulus at most $n^{r}$ for $n \geq r$ , so $E_{P_{n}} [p_{ρ}^{#}] = O (n^{∣ ρ ∣})$ uniformly in $n$ ; and if $m_{1} (ρ) = 0$ with $ρ \neq \emptyset$ then $ρ \neq (1^{r})$ , so the expectation vanishes for every $n$ , as asserted.

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The shifted character observables form a basis of $A$ , with the Kerov weight filtration

Statement

Let $A$ be the commutative $R$ -algebra of functions on $D_{0}$ generated by the profile moments ${\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots$ of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ , so $A = R [{\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots]$ as a polynomial algebra, and regard its elements as functions on Young diagrams through $λ \mapsto λ (\cdot)$ . Let $p_{ρ}^{#}$ be the shifted character observables of the same item, $∣ ρ ∣_{1} = ∣ ρ ∣ + m_{1} (ρ)$ and $wt ({\tilde{p}}_{k}) : = k$ . Then:

(i) every function $p_{ρ}^{#}$ belongs to $A$ , and the family ${p_{ρ}^{#}}_{ρ}$ is a linear basis of $A$ ;

(ii) $\deg_{1} (p_{ρ}^{#}) : = ∣ ρ ∣_{1}$ defines an algebra filtration: if $p_{σ}^{#} p_{τ}^{#} = \sum_{ρ} f_{σ τ}^{ρ} p_{ρ}^{#}$ then $f_{σ τ}^{ρ} \neq 0$ implies $\deg_{1} (p_{ρ}^{#}) \leq \deg_{1} (p_{σ}^{#}) + \deg_{1} (p_{τ}^{#})$ ; moreover the weights $wt (p_{ρ}^{#}) : = ∣ ρ ∣ + ℓ (ρ)$ define an algebra filtration which dominates the first one, $\deg_{1} (f) \leq wt (f)$ for every $f \in A$ , and which is the weight filtration generated by $wt ({\tilde{p}}_{k}) = k$ ;

(iii) the top term with respect to the weight filtration is unique with coefficient one: $p_{σ}^{#} p_{τ}^{#} = p_{σ \cup τ}^{#} + ⟨ terms of strictly smaller weight ⟩,$ where $σ \cup τ$ is the partition obtained by uniting all parts of $σ$ and $τ$ .

In particular, the monomials in ${\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots$ (including $1$ ) and the family ${p_{ρ}^{#}}$ are two linear bases of the same algebra. Their change of basis is triangular in canonical degree, where $\deg (p_{j}) = j$ , and compatible with the weight filtration. The two filtrations in (ii) are distinct: for $p_{(2)}^{#} p_{(2)}^{#} = p_{(2, 2)}^{#} + 4 p_{(3)}^{#} + 2 p_{(1, 1)}^{#}$ the term $2 p_{(1, 1)}^{#}$ still has $\deg_{1} = 4 = \deg_{1} (p_{(2)}^{#}) + \deg_{1} (p_{(2)}^{#})$ , so the unique-top-term statement (iii) is asserted for the weight filtration (there $2 p_{(1, 1)}^{#}$ has weight $4 < 6$ ). The source's claim " $∣ ρ ∣_{N} = ∣ ρ ∣ + ℓ (ρ)$ filtration equals the weight filtration" is the corresponding statement for $J = N$ , not for $J = {1}$ .

Facts & Assumptions

Given: partitions $ρ, σ, τ$ ; the observables $p_{ρ}^{#}$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ); the partial-permutation algebra $A_{\infty}$ of [IK] with basis ${A_{ρ}}$ and structure constants $g_{σ τ}^{ρ}$ , $A_{σ} A_{τ} = \sum_{ρ} g_{σ τ}^{ρ} A_{ρ}$ ; and $z_{ρ} = \prod_{k} k^{m_{k} (ρ)} m_{k} (ρ)!$ .

[F1]

(IK Prop. 6.2, Prop. 6.3, Remark 6.4 — imported with the locators recorded above.) For a fixed permutation $w_{ρ}$ of cycle type $ρ$ on a set $d_{ρ}$ of size $∣ ρ ∣$ , the coefficient $g_{σ τ}^{ρ}$ equals the number of pairs $((d_{1}, w_{1}), (d_{2}, w_{2}))$ of partial permutations with $d_{1} \cup d_{2} = d_{ρ}$ and $w_{1} w_{2} = w_{ρ}$ , $w_{i}$ of cycle structure $σ, τ$ on $d_{i}$ . Consequently $g_{σ τ}^{ρ} \neq 0$ implies $∣ ρ ∣ \leq ∣ σ ∣ + ∣ τ ∣$ ; the unique partition with $∣ ρ ∣ = ∣ σ ∣ + ∣ τ ∣$ and $g_{σ τ}^{ρ} \neq 0$ is $ρ = σ \cup τ$ , and $g_{σ τ}^{σ \cup τ} = \prod_{k \geq 1} (\binom{m_{k} (σ) + m_{k} (τ)}{m_{k} (σ)})$ .

[F2]

(IK Thm. 9.1 and (9.3) — imported.) The linear map $F (A_{ρ}) = p_{ρ}^{#} / z_{ρ}$ is an isomorphism of algebras onto the algebra of shifted symmetric functions, where $p_{ρ}^{#} (λ) = n^{↓ ∣ ρ ∣} χ_{ρ \cup 1^{n - ∣ ρ ∣}}^{λ} / \dim_{C} S^{λ}$ for $n \geq ∣ ρ ∣$ ; hence the functions $p_{ρ}^{#}$ are linearly independent and their structure constants satisfy $f_{σ τ}^{ρ} = \frac{z_{σ} z_{τ}}{z_{ρ}} g_{σ τ}^{ρ}$ (IvOl Prop. 4.5). In particular $f_{σ τ}^{σ \cup τ} = 1$ (IvOl formula (4.1)), since $\frac{z_{σ} z_{τ}}{z_{σ \cup τ}}$ is the reciprocal of the binomial product of [F1].

[F3]

(IvOl Prop. 4.2 and Cor. 4.3 — imported.) Each $p_{ρ}^{#}$ belongs to the algebra generated by the canonical power sums, whose top homogeneous component is $p_{ρ}$ , and the ${p_{ρ}^{#}}$ form a basis of that algebra; this algebra is freely generated by the profile moments restricted to Young diagrams, by IvOl Proposition 1.5, Proposition 2.7 and Corollary 2.8 (printed pp. 8, 13-14). Specifically, ${\tilde{p}}_{k} = k p_{k - 1}$ plus a linear combination of $p_{k - 2}, \dots, p_{1}$ , where $p_{j} (λ) = \sum_{i \geq 1} ((λ_{i} - i + 1 / 2)^{j} - (- i + 1 / 2)^{j})$ (zeros are appended to $λ$ ).

[F4]

(IvOl Prop. 4.7, Cor. 4.8, Prop. 4.9, Prop. 4.10 — imported.) For every $J \subseteq N$ the degrees $\deg_{J} (p_{ρ}^{#}) = ∣ ρ ∣ + \sum_{j \in J} m_{j} (ρ)$ satisfy the inequality $f_{σ τ}^{ρ} \neq 0 \Rightarrow \deg_{J} (p_{ρ}^{#}) \leq \deg_{J} (p_{σ}^{#}) + \deg_{J} (p_{τ}^{#})$ , so they define algebra filtrations; in the equality case $\deg_{J} (p_{ρ}^{#}) = \deg_{J} (p_{σ}^{#}) + \deg_{J} (p_{τ}^{#})$ the counting argument of IK gives the structural constraint recorded in Cor. 4.8. For $J = N$ this yields $p_{σ}^{#} p_{τ}^{#} = p_{σ \cup τ}^{#} + (lower \deg_{N} terms)$ , and the $\deg_{N}$ filtration coincides with the weight filtration generated by $wt ({\tilde{p}}_{k}) = k$ .

Proof

technique · direct

1.1givenF3algebra

Membership and basis: let $A_{Y}$ be the source's algebra of functions on Young diagrams in [F3]. Its profile-moment generators are algebraically independent by [F3]. Restriction from the algebra $A$ of profile polynomials on $D_{0}$ onto $A_{Y}$ is surjective, and is injective: a polynomial that vanishes on $D_{0}$ vanishes on all Young profiles, so independence in $A_{Y}$ forces its coefficients to be zero; likewise a function in $A$ vanishing on all Young profiles is the zero polynomial. Hence restriction is an isomorphism. Each $p_{ρ}^{#}$ has a unique polynomial extension to $D_{0}$ , and [F3] gives the linear basis of these extensions. The empty partition labels $p_{\emptyset}^{#} = 1$ . This proves (i), including that $A$ is a polynomial algebra.

2.1givenF2F4step 1.1

Structure constants and the two filtrations: by [F2] the structure constants of $A$ in the basis ${p_{ρ}^{#}}$ are $f_{σ τ}^{ρ} = \frac{z_{σ} z_{τ}}{z_{ρ}} g_{σ τ}^{ρ}$ , and by [F4] the degrees $\deg_{J}$ are compatible with multiplication; taking $J = {1}$ gives $\deg_{1} (p_{ρ}^{#}) \leq \deg_{1} (p_{σ}^{#}) + \deg_{1} (p_{τ}^{#})$ whenever $f_{σ τ}^{ρ} \neq 0$ . Taking $J = N$ gives weight compatibility, and [F4] identifies the $\deg_{N}$ filtration with the weight filtration $wt ({\tilde{p}}_{k}) = k$ , so $wt (p_{ρ}^{#}) = ∣ ρ ∣ + ℓ (ρ)$ and $wt$ is an algebra filtration. Since $m_{1} (ρ) \leq ℓ (ρ)$ , $\deg_{1} \leq wt$ on each basis element, hence on $A$ . This is (ii).

3.1givenF1F2F4step 2.1

Unique top term: by [F1] the only partition with $∣ ρ ∣ = ∣ σ ∣ + ∣ τ ∣$ and $g_{σ τ}^{ρ} \neq 0$ is $ρ = σ \cup τ$ , and by [F2] its coefficient is $f_{σ τ}^{σ \cup τ} = 1$ . Any other contributing $ρ$ satisfies $\deg_{N} (p_{ρ}^{#}) < \deg_{N} (p_{σ \cup τ}^{#})$ : otherwise, if the $\deg_{N}$ -degrees were equal, the equality case recorded in [F4] (with $J = N$ ) would force $ρ = σ \cup τ$ by the same imported argument. Since $\deg_{N}$ is the weight, all other terms have strictly smaller weight. Hence $p_{σ}^{#} p_{τ}^{#} = p_{σ \cup τ}^{#} + (lower weight)$ , which is (iii).

4.1givenF3F4step 1.1step 2.1algebra∎

Triangularity: [F3] gives ${\tilde{p}}_{k} = k p_{k - 1}$ plus lower canonical degree and $p_{ρ}^{#} = p_{ρ}$ plus lower canonical degree, where $p_{ρ} = \prod_{i} p_{ρ_{i}}$ . Thus the monomial $\prod_{i} {\tilde{p}}_{ρ_{i} + 1}$ has leading canonical component $\prod_{i} (ρ_{i} + 1) p_{ρ}$ , so its expansion in the shifted-character basis is triangular with nonzero diagonal. These monomials, including the empty product, form a basis because the generators are algebraically independent. The equality of weight filtrations in [F4] makes this change compatible with weight levels. The generators themselves are algebra generators, rather than a linear basis. This proves the final assertion. No choice principle is used in the finite algebraic operations or cited algebraic results.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Normalized shifted character observables $η_{ρ}$

Definition

Fix $n \geq 1$ . For a partition $ρ$ of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ write $∣ ρ ∣_{1} : = ∣ ρ ∣ + m_{1} (ρ)$ , where $m_{1} (ρ)$ is the multiplicity of the part $1$ . For $λ ⊢ n$ with $n \geq ∣ ρ ∣$ define the normalized observable $η_{ρ}^{(n)} (λ) : = \frac{p_{ρ}^{#} (λ)}{n^{∣ ρ ∣_{1} / 2} \prod_{k \geq 2} k^{m_{k} (ρ) / 2}} .$ For $n < ∣ ρ ∣$ declare $η_{ρ}^{(n)} (λ) : = 0$ , consistently with $p_{ρ}^{#} (λ) = 0$ there. Since $p_{1}^{#} (λ) = n$ , the definition can be written in the equivalent localized form $η_{ρ}^{(n)} = \frac{p_{ρ}^{#}}{(p_{1}^{#})^{m_{1} (ρ)} \prod_{k \geq 2} (k (p_{1}^{#})^{k})^{m_{k} (ρ) / 2}} on Y_{n},$ which is the concrete evaluation of the source's localization $A_{e x t} = A [(p_{1}^{#})^{1 / 2}, (p_{1}^{#})^{- 1 / 2}]$ on each $Y_{n}$ ; the equality uses $p_{1}^{#} (λ) = n$ , a positive number for $n \geq 1$ , so the square roots are ordinary positive real roots. In particular, for a single part $ρ = (k)$ , $k \geq 2$ , one has $η_{(k)}^{(n)} = p_{k}^{#} / (\sqrt{k} n^{k / 2}) = η_{k}^{(n)}$ , the observable of Joint convergence in distribution and the normalized cycle-character observables.

Each $η_{ρ}^{(n)}$ is a real function on the finite set $Y_{n}$ , hence a random variable on $(Y_{n}, P_{n})$ (The Plancherel measure on the partitions of $n$ ), and for every $n \geq ∣ ρ ∣$ the expectation of Plancherel expectations of the shifted character observables gives $E_{P_{n}} [η_{ρ}^{(n)}] = {\begin{cases} \frac{n^{↓ ∣ ρ ∣}}{n^{∣ ρ ∣_{1} / 2} \prod_{k \geq 2} k^{m_{k} (ρ) / 2}}, & ρ = (1^{∣ ρ ∣}), \\ 0, & ρ \neq (1^{∣ ρ ∣}), \end{cases}$ so the expectation vanishes whenever $m_{1} (ρ) = 0$ and $ρ \neq \emptyset$ , and equals $\prod_{j = 0}^{∣ ρ ∣ - 1} (1 - j / n) \leq 1$ for $ρ = (1^{∣ ρ ∣})$ ; in every case it is $O (1)$ uniformly in $n$ . Moreover $∣ η_{ρ}^{(n)} (λ) ∣ \leq n^{(∣ ρ ∣ - m_{1} (ρ)) / 2} \prod_{k \geq 2} k^{- m_{k} (ρ) / 2}$ for every $λ ⊢ n$ , $n \geq ∣ ρ ∣$ : by For a complex character, $χ (1) = \dim V$ , $χ$ is a class function, and $∣ χ (g) ∣ \leq χ (1)$ with equality exactly at scalars every irreducible character satisfies $∣ χ_{μ}^{λ} ∣ \leq χ_{(1^{n})}^{λ} = \dim_{C} S^{λ}$ , so $∣ p_{ρ}^{#} (λ) ∣ \leq n^{↓ ∣ ρ ∣} \leq n^{∣ ρ ∣}$ , and dividing by $n^{∣ ρ ∣_{1} / 2} \prod_{k \geq 2} k^{m_{k} (ρ) / 2} = n^{(∣ ρ ∣ + m_{1} (ρ)) / 2} \prod_{k \geq 2} k^{m_{k} (ρ) / 2}$ gives the displayed bound. This normalization is the one used in Hermite leading terms for normalized shifted characters and Kerov's central limit theorem for normalized cycle characters. No choice principle is used.

LemmaStatement: Literature-sourcedProof: Literature-sourcedOpen item page →

The profile-moment generators in the shifted-character basis

Statement

Let $B (u) : = 1 + \sum_{j \geq 2} p_{j - 1}^{#} u^{j}$ be the formal power series with coefficients in $A$ . Then for every $k \geq 2$ ${\tilde{p}}_{k} = [u^{k}] (B (u)^{k}) + ⟨ terms of weight \leq k - 1 ⟩,$ the top weight component of ${\tilde{p}}_{k}$ being exactly the weight- $k$ component of the coefficient of $u^{k}$ in $B (u)^{k}$ , on which $p_{k - 1}^{#}$ occurs with coefficient $k$ . Consequently the linear functionals $L_{k} (p_{ρ}^{#}) : = {\begin{cases} 1, & k even and ρ = (1^{k / 2}), \\ 0, & otherwise, \end{cases}$ defined by linear extension on the full basis of $A$ (with $L_{0} (1) = 1$ ), then restricted to weight-homogeneous elements of weight $k$ , are multiplicative: if $f$ and $g$ are weight-homogeneous of weights $k^{'}$ and $k - k^{'}$ , then $L_{k} (f g) = L_{k^{'}} (f) L_{k - k^{'}} (g)$ ; and $L_{2 m} ({\tilde{p}}_{2 m}) = (\binom{2 m}{m}), L_{k} ({\tilde{p}}_{k}) = 0 (k odd) .$

Facts & Assumptions

Given: the algebra $A = R [{\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots]$ with the observables $p_{ρ}^{#}$ and the profile moments ${\tilde{p}}_{k} = k (k - 1) \int x^{k - 2} σ_{ω} d x$ of Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ , and the formal series $B (u) = 1 + \sum_{j \geq 2} p_{j - 1}^{#} u^{j}$ .

[F1]

(IvOl Prop. 3.7, imported with the locator above.) For every $k \geq 2$ , ${\tilde{p}}_{k} = [u^{k}] {B (u)^{k}}$ plus a polynomial in $p_{1}^{#}, \dots, p_{k - 2}^{#}$ of total weight at most $k - 1$ ; this is the inversion of the top-weight relation of IvOl Prop. 3.5, obtained there by Lagrange inversion.

[F2]

The weight filtration and top-term rule: the weights $wt (p_{ρ}^{#}) = ∣ ρ ∣ + ℓ (ρ)$ define an algebra filtration coinciding with $wt ({\tilde{p}}_{k}) = k$ , and $p_{σ}^{#} p_{τ}^{#} = p_{σ \cup τ}^{#} + (lower weight)$ (The shifted character observables form a basis of $A$ , with the Kerov weight filtration).

[F3]

$\deg_{1} (p_{ρ}^{#}) = ∣ ρ ∣ + m_{1} (ρ) \leq wt (p_{ρ}^{#})$ , and $\deg_{1}$ is compatible with multiplication (The shifted character observables form a basis of $A$ , with the Kerov weight filtration). The top-term rule applied repeatedly also gives $(p_{1}^{#})^{m} = p_{(1^{m})}^{#} + (lower weight)$ .

Proof

technique · direct

1.1givenF1F2F3

Expansion: by [F1], ${\tilde{p}}_{k} = [u^{k}] {B (u)^{k}} + (terms of weight \leq k - 1)$ . Every product appearing in $[u^{k}] {B (u)^{k}}$ has the form $p_{j_{1} - 1}^{#} \dots p_{j_{i} - 1}^{#}$ with $j_{1} + \dots + j_{i} = k$ and total weight $j_{1} + \dots + j_{i} = k$ , and by the top-term rule of [F2] its weight- $k$ component is $p_{(j_{1} - 1) \cup \dots \cup (j_{i} - 1)}^{#}$ with coefficient $1$ ; in particular the term with a single factor ( $i = 1$ , $j_{1} = k$ ) contributes $k p_{k - 1}^{#}$ , so $p_{k - 1}^{#}$ occurs in the top weight component of ${\tilde{p}}_{k}$ with coefficient $k$ . Hence the top weight component of ${\tilde{p}}_{k}$ is exactly the weight- $k$ component of $[u^{k}] {B (u)^{k}}$ .

2.1givenF2F3step 1.1algebra

Multiplicativity of $L_{k}$ : if $k$ is odd, $L_{k} = 0$ and at least one of $L_{k^{'}}$ , $L_{k - k^{'}}$ is zero, proving the identity. Suppose $k$ is even; let $f, g$ be weight-homogeneous of weights $k^{'}$ and $k - k^{'}$ and expand them in the basis ${p_{σ}^{#}}$ , which is possible by The shifted character observables form a basis of $A$ , with the Kerov weight filtration(i). Since the weight filtration has level $r$ spanned by the $p_{σ}^{#}$ with $wt (p_{σ}^{#}) \leq r$ ([F2]), only $σ$ with $wt (p_{σ}^{#}) \leq k^{'}$ and $τ$ with $wt (p_{τ}^{#}) \leq k - k^{'}$ occur. The coefficient of $p_{(1^{k / 2})}^{#}$ in $f g$ is $\sum_{σ, τ} f_{σ} g_{τ} f_{σ τ}^{(1^{k / 2})}$ ; by [F3] and [F2], $f_{σ τ}^{ρ} \neq 0$ forces $\deg_{1} (p_{ρ}^{#}) \leq \deg_{1} (p_{σ}^{#}) + \deg_{1} (p_{τ}^{#}) \leq wt (p_{σ}^{#}) + wt (p_{τ}^{#}) \leq k$ , so a nonzero contribution to $ρ = (1^{k / 2})$ , where $\deg_{1} (p_{ρ}^{#}) = k$ , forces equalities $\deg_{1} (p_{σ}^{#}) = wt (p_{σ}^{#}) = k^{'}$ and $\deg_{1} (p_{τ}^{#}) = wt (p_{τ}^{#}) = k - k^{'}$ ; as $\deg_{1} \leq wt$ with equality only for columns, this forces $σ = (1^{k^{'} / 2})$ and $τ = (1^{(k - k^{'}) / 2)}$ , so $k$ and $k^{'}$ are even and the coefficient is $f_{σ} g_{τ} \cdot 1$ (the top coefficient being $1$ by [F2]). Summing gives $L_{k} (f g) = L_{k^{'}} (f) L_{k - k^{'}} (g)$ ; when $k$ or $k^{'}$ is odd, no such pair exists and both sides are $0$ .

2.2givenF2step 1.1algebra

Values on the generators: expand $[u^{2 m}] B (u)^{2 m}$ as a finite sum of products $p_{j_{1} - 1}^{#} \dots p_{j_{i} - 1}^{#}$ with $j_{l} \geq 2$ and $\sum_{l} j_{l} = 2 m$ . By [F2], the only weight- $2 m$ partition in such a product is $(j_{1} - 1) \cup \dots \cup (j_{i} - 1)$ , with coefficient one; lower-weight terms cannot contribute to $L_{2 m}$ . This partition is $(1^{m})$ precisely when $i = m$ and all $j_{l} = 2$ . There are $(\binom{2 m}{m})$ ways to select the $m$ factors supplying $p_{1}^{#} u^{2}$ among the $2 m$ factors of $B (u)^{2 m}$ . Consequently $L_{2 m} ({\tilde{p}}_{2 m}) = (\binom{2 m}{m})$ by step 1.1. For odd $k$ , $L_{k}$ is zero by definition.

3.1givenstep 1.1step 2.1step 2.2∎

Conclusion: step 1.1 gives the stated expansion with its equivalent form and the description of the top weight component; step 2.1 gives multiplicativity of $L_{k}$ ; step 2.2 evaluates $L$ on the generators ${\tilde{p}}_{k}$ as the central binomial coefficients for even $k$ and $0$ for odd $k$ . No choice principle is used: the imported IvOl statements are algebraic.

LemmaStatement: Literature-sourcedProof: Literature-sourcedOpen item page →

Shifted character products: exact for $p_{1}^{#}$ and leading terms for $p_{k}^{#}$

Statement

For every partition $σ$ :

(i) (exact, all orders) $p_{σ}^{#} p_{1}^{#} = p_{σ \cup 1}^{#} + ∣ σ ∣ p_{σ}^{#}$ ;

(ii) for every $k \geq 2$ , $p_{σ}^{#} p_{k}^{#} = p_{σ \cup k}^{#} + {\begin{cases} k m_{k} (σ) p_{(σ ∖ k) \cup 1^{k}}^{#}, & m_{k} (σ) \geq 1, \\ 0, & m_{k} (σ) = 0, \end{cases} + ⟨ terms of strictly smaller \deg_{1} ⟩,$ where $σ ∖ k$ removes one part equal to $k$ .

In particular, for $m \geq 1$ , $k \geq 2$ , $p_{(k^{m})}^{#} p_{k}^{#} = p_{(k^{m + 1})}^{#} + k m p_{(k^{m - 1}, 1^{k})}^{#} + ⟨ lower \deg_{1} terms ⟩,$ which is the recurrence behind the Hermite leading-term lemma. All exponents $σ \cup k$ , $(σ ∖ k) \cup 1^{k}$ are partitions in the sense of Partitions, English diagrams, and conjugation.

Facts & Assumptions

Given: partitions $σ, τ$ and $k \geq 2$ ; the observables $p_{ρ}^{#}$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ); the algebra $A$ with the basis ${p_{ρ}^{#}}$ , the structure constants $f_{σ τ}^{ρ}$ , the filtration $\deg_{1} (p_{ρ}^{#}) = ∣ ρ ∣ + m_{1} (ρ)$ , and the top-term rule (The shifted character observables form a basis of $A$ , with the Kerov weight filtration); $z_{ρ} = \prod_{k} k^{m_{k} (ρ)} m_{k} (ρ)!$ and the partial-permutation structure constants $g_{σ τ}^{ρ}$ .

[F1]

(i) For $∣ σ ∣ = r$ and $n \geq r + 1$ one has $p_{σ}^{#} (λ) = n^{↓ r} χ_{σ \cup 1^{n - r}}^{λ} / \dim λ$ and $p_{σ \cup 1}^{#} (λ) = n^{↓ (r + 1)} χ_{σ \cup 1^{n - r}}^{λ} / \dim λ$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ ); the falling factorial satisfies $n^{↓ r} \cdot n = n^{↓ (r + 1)} + n^{↓ r} r$ (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F2]

(ii) Structure constants: $f_{σ τ}^{ρ} = \frac{z_{σ} z_{τ}}{z_{ρ}} g_{σ τ}^{ρ}$ and $f_{σ τ}^{σ \cup τ} = 1$ (The shifted character observables form a basis of $A$ , with the Kerov weight filtration). For the degree-one equality case, IvOl Corollary 4.8 (of the proof of Proposition 4.7), with $J = {1}$ , says no fixed point of $s_{1}$ or $s_{2}$ lies in $X_{1} \cap X_{2}$ , while every point of $X_{1} \cap X_{2}$ is fixed by $s = s_{1} s_{2}$ . Combined with the partial-permutation count in IK Proposition 6.2, this forces the overlap description used in step 1.2: it is a union of common nontrivial cycles of $s_{1}$ and $s_{2}^{- 1}$ . The structure constants and the exact Corollary 4.8 locator are recorded in the source references above.

Proof

technique · direct

1.1givenF1algebra

Exact product with $p_{1}^{#}$ : fix $λ ⊢ n$ with $n \geq ∣ σ ∣ + 1$ (for $n < ∣ σ ∣$ both sides vanish; at $n = ∣ σ ∣$ one has $p_{σ \cup 1}^{#} = 0$ and $p_{1}^{#} = ∣ σ ∣$ , so the identity holds directly). By [F1], $p_{σ}^{#} (λ) p_{1}^{#} (λ) = n^{↓ r} χ_{σ \cup 1^{n - r}}^{λ} / \dim λ \cdot n$ and $p_{σ \cup 1}^{#} (λ) = n^{↓ (r + 1)} χ_{σ \cup 1^{n - r}}^{λ} / \dim λ$ , so the claim reduces to $n^{↓ r} \cdot n = n^{↓ (r + 1)} + n^{↓ r} r$ , which is the defining recursion $n^{↓ (r + 1)} = n^{↓ r} (n - r)$ rewritten; hence $p_{σ}^{#} p_{1}^{#} = p_{σ \cup 1}^{#} + ∣ σ ∣ p_{σ}^{#}$ .

1.2givenF2algebra

Equality-case analysis: let $ρ$ be such that $f_{σ (k)}^{ρ} \neq 0$ and $\deg_{1} (p_{ρ}^{#}) = \deg_{1} (p_{σ}^{#}) + k$ , where $τ = (k)$ and $\deg_{1} (p_{k}^{#}) = k$ because $m_{1} ((k)) = 0$ for $k \geq 2$ . By [F2] the equality case forces the overlap $X_{1} \cap X_{2}$ to consist of common nontrivial cycles of $s_{1}$ and $s_{2}^{- 1}$ ; since $s_{2}$ is a single $k$ -cycle, either $X_{1} \cap X_{2} = \emptyset$ , giving $ρ = σ \cup k$ with coefficient $f_{σ (k)}^{σ \cup k} = 1$ , or $X_{1} \cap X_{2}$ is one common $k$ -cycle, which requires $m_{k} (σ) \geq 1$ , gives $ρ = (σ ∖ k) \cup 1^{k}$ , and forces $X_{2} \subseteq X_{1}$ .

1.3givenF2algebra

Coefficient in the second case: in the case $ρ = (σ ∖ k) \cup 1^{k}$ abbreviate $m : = m_{k} (σ) \geq 1$ and $ℓ : = m_{1} (σ)$ ; a direct computation from $z_{μ} = \prod_{i} i^{m_{i} (μ)} m_{i} (μ)!$ gives $\frac{z_{ρ}}{z_{σ} z_{(k)}} = \frac{(k + ℓ)!}{ℓ! k^{2} m}$ , so by [F2] the identity $f_{σ (k)}^{ρ} = k m$ is equivalent to $g_{σ (k)}^{ρ} = \frac{(k + ℓ)!}{ℓ! k}$ . The partial-permutation count recorded in [F2] in this case is the number of ways to choose a $k$ -cycle inside the $(k + ℓ)$ -point fixed-point set of $w_{ρ}$ : all other cycles of $w_{ρ}$ must remain unchanged: choose the $k$ -point support, $(\binom{k + ℓ}{k}) = \frac{(k + ℓ)!}{k! ℓ!}$ ways, and a $k$ -cycle on it, $(k - 1)!$ ways, giving $g_{σ (k)}^{ρ} = \frac{(k + ℓ)!}{ℓ! k}$ as required; hence $f_{σ (k)}^{ρ} = k m_{k} (σ)$ , the factor $m = m_{k} (σ)$ counting the choice of which $k$ -part of $σ$ is the common cycle.

2.1givenstep 1.1step 1.2step 1.3∎

Conclusion: every other contributing $ρ$ has $\deg_{1} (p_{ρ}^{#}) < \deg_{1} (p_{σ}^{#}) + k$ by the definition of the equality case in step 1.2, so the expansion takes the displayed form; specialising $σ = (k^{m})$ gives $m_{k} (σ) = m$ and $(σ ∖ k) \cup 1^{k} = (k^{m - 1}, 1^{k})$ , which is the stated recurrence. No choice principle is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Hermite leading terms for normalized shifted characters

Statement

For every partition $ρ$ with $m_{1} (ρ) = 0$ and every $n \geq \max {1, ∣ ρ ∣}$ , the normalized observables satisfy $\prod_{k \geq 2} H_{m_{k} (ρ)} (η_{k}^{(n)}) = η_{ρ}^{(n)} + R_{ρ}^{(n)} on Y_{n},$ where the remainder admits a finite expansion $R_{ρ}^{(n)} = \sum_{σ, j} c_{σ, j} n^{- j / 2} η_{σ}^{(n)}$ with real constants $c_{σ, j}$ and $j \geq 1$ such that the total degree $∣ σ ∣_{1} - j$ is strictly smaller than $∣ ρ ∣_{1}$ , and $σ$ runs over partitions with $m_{1} (σ) = 0$ . Consequently $∣ E_{P_{n}} [\prod_{k \geq 2} H_{m_{k} (ρ)} (η_{k}^{(n)})] - E_{P_{n}} [η_{ρ}^{(n)}] ∣ = O (n^{- 1 / 2}) .$ In particular, if $ρ \neq \emptyset$ then the expectation of the Hermite product is $O (n^{- 1 / 2})$ , and if $ρ = \emptyset$ it is $1$ .

Facts & Assumptions

Given: a partition $ρ$ with $m_{1} (ρ) = 0$ ; the observables $η_{σ}^{(n)} = p_{σ}^{#} / (n^{∣ σ ∣_{1} / 2} \prod_{k \geq 2} k^{m_{k} (σ) / 2})$ of Normalized shifted character observables $η_{ρ}$ and the Hermite polynomials $H_{m}$ of The monic probabilists' Hermite polynomials.

[F1]

The degrees $\deg_{1} (p_{τ}^{#}) = ∣ τ ∣ + m_{1} (τ)$ form an algebra filtration, and the partial-permutation structure constants count pairs on supports whose union has size $∣ ν ∣ \leq ∣ σ ∣ + ∣ τ ∣$ (The shifted character observables form a basis of $A$ , with the Kerov weight filtration). The exact identities and the single-cycle top-degree expansion are Shifted character products: exact for $p_{1}^{#}$ and leading terms for $p_{k}^{#}$ . Its equality-case argument also gives the distinct-size rule: if $σ, τ$ have no common part, then $p_{σ}^{#} p_{τ}^{#} = p_{σ \cup τ}^{#} + (lower \deg_{1})$ , because an overlap of equal degree must consist of common nontrivial cycles, impossible here. This is Ivanov--Olshanski Corollary 4.13, printed p. 25, whose full proof is the same support-count argument. No arbitrary unique-top-term rule is asserted for $\deg_{1}$ .

[F2]

Hermite recurrence: $x H_{m} (x) = H_{m + 1} (x) + m H_{m - 1} (x)$ with $H_{0} = 1$ , $H_{1} = x$ (The monic probabilists' Hermite polynomials).

[F3]

Expectations: $E_{P_{n}} [η_{σ}^{(n)}] = 0$ whenever $m_{1} (σ) = 0$ , $σ \neq \emptyset$ , and $E_{P_{n}} [η_{σ}^{(n)}] = O (1)$ uniformly in $n$ in every case (Normalized shifted character observables $η_{ρ}$ , Plancherel expectations of the shifted character observables).

Proof

technique · direct

1.1F1givenalgebra

Exact removal of ones. For any partition $τ$ with no ones and $q \geq 0$ , repeated use of the exact $p_{1}^{#}$ identity in [F1], together with $p_{1}^{#} = n$ , gives $p_{τ \cup 1^{q}}^{#} = p_{τ}^{#} \prod_{h = 0}^{q - 1} (n - ∣ τ ∣ - h)$ on every $Y_{n}$ with $n \geq 1$ . Dividing by the normalization gives $η_{τ \cup 1^{q}}^{(n)} = η_{τ}^{(n)} \prod_{h = 0}^{q - 1} (1 - (∣ τ ∣ + h) / n)$ . Thus every normalized term with ones is a finite polynomial in $1 / n$ times the corresponding observable without ones; its constant coefficient is one. These identities also hold below the partition size, because either $p_{τ}^{#} = 0$ or the product contains a zero factor.

2.1F1step 1.1algebra

Negative-degree remainders. Write a term as $c n^{a / 2} p_{ν}^{#}$ and assign it degree $a + ∣ ν ∣_{1}$ . The algebra-filtration inequality in [F1] makes degrees subadditive under multiplication, and $n = p_{1}^{#}$ has degree two. Each $η_{ν}$ has degree zero. If a term has negative degree, dividing by the normalization rewrites it as $c^{'} n^{- j / 2} η_{ν}$ with an integer $j \geq 1$ . Removing its ones by step 1.1 produces finitely many terms $c^{''} n^{- j^{'} / 2} η_{τ}$ with $j^{'} \geq j \geq 1$ and no ones in $τ$ . This rule is algebraic and exact, rather than a pointwise bound on the observables.

3.1F1F2step 1.1step 2.1algebra

A single cycle size. Fix $k \geq 2$ and put $f_{m} = η_{(k^{m})}$ , with $f_{0} = 1$ and $f_{1} = η_{k}$ . Divide the single-cycle multiplication formula of [F1] by ${\sqrt{k}}^{m + 1} n^{k (m + 1) / 2}$ . It gives $f_{m} η_{k} = f_{m + 1} + m η_{(k^{m - 1}, 1^{k})} + r_{m}$ , where $r_{m}$ has negative degree. Step 1.1 replaces the middle observable by $f_{m - 1}$ plus negative-degree terms, so $f_{m} η_{k} = f_{m + 1} + m f_{m - 1} + r_{m}^{'}$ with $r_{m}^{'}$ of negative degree. Comparing with the recurrence [F2] proves by induction $f_{m} = H_{m} (η_{k}) + E_{m}$ , where $E_{0} = E_{1} = 0$ and $E_{m + 1} = η_{k} E_{m} - m E_{m - 1} - r_{m}^{'}$ has negative degree by step 2.1. In particular the contraction coefficient is $m$ ; no extra power of $n$ remains.

4.1F1step 1.1step 2.1step 3.1algebra

Combining distinct sizes. Group the parts of $ρ$ into the blocks $(k^{m_{k} (ρ)})$ with distinct $k$ . Successive applications of the distinct-size rule in [F1], with normalization denominators multiplying exactly, give $\prod_{k} η_{(k^{m_{k}})} = η_{ρ} + (negative-degree terms)$ . Substituting step 3.1 and expanding the finite product, every correction includes a negative-degree $E_{m}$ and other factors of degree at most zero. Therefore $\prod_{k} H_{m_{k}} (η_{k}) = η_{ρ} + R_{ρ}$ with $R_{ρ}$ of negative degree. Step 2.1 writes it exactly as a finite sum $\sum_{σ, j} c_{σ, j} n^{- j / 2} η_{σ}$ with $j \geq 1$ and $m_{1} (σ) = 0$ . The support-union bound in [F1] shows that every partition in a product of cycle observables has size at most the sum of the cycle sizes; every Hermite monomial has that sum at most $∣ ρ ∣$ . Removing ones only decreases size, so $∣ σ ∣ \leq ∣ ρ ∣$ , and hence $∣ σ ∣_{1} - j < ∣ ρ ∣_{1}$ . All constants are independent of $n$ .

5.1F3step 4.1algebra∎

Expectations. The finite remainder expansion in step 4.1 and [F3] give $∣ E [R_{ρ}^{(n)}] ∣ = O (n^{- 1 / 2})$ : each term has $j \geq 1$ and uniformly bounded expectation, indeed zero for nonempty $σ$ without ones. For nonempty $ρ$ its own expectation is zero by [F3], so the Hermite-product expectation is $O (n^{- 1 / 2})$ . For $ρ = \emptyset$ both empty products equal one and the remainder vanishes. This proves the exact expansion and all stated consequences.

PropositionStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Scaled Plancherel profile moments converge in probability

Statement

For $n \geq 1$ , let $λ$ range over $Y_{n}$ under the Plancherel measures $P_{n}$ , let $\overset{ˉ}{λ}$ be the $\sqrt{n}$ -scaled Russian profile of Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram, and let $Ω$ be the limit profile of The Logan-Shepp-Vershik-Kerov limit profile $Ω$ . Under the Plancherel measures $P_{n}$ , for every integer $k \geq 0$ , $\int_{R} (\overset{ˉ}{λ} (x) - Ω (x)) x^{k} d x ⟶ 0 in probability as n \to \infty .$ Equivalently, ${\tilde{p}}_{j} [\overset{ˉ}{λ}] \to {\tilde{p}}_{j} [Ω]$ in probability for every $j \geq 2$ . Moreover, for every $f$ in the algebra $A = R [{\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots]$ , $E_{P_{n}} [f (\overset{ˉ}{λ})] ⟶ f [Ω] .$

Facts & Assumptions

Given: $n \geq 1$ ; the probability space $(Y_{n}, P_{n})$ of The Plancherel measure on the partitions of $n$ and The Plancherel weights sum to one; the observables $p_{ρ}^{#}$ and the profile moments ${\tilde{p}}_{k} [ω] = k (k - 1) \int_{R} x^{k - 2} σ_{ω} (x) d x$ with $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ , together with the scaling identity ${\tilde{p}}_{k} [\overset{ˉ}{λ}] = n^{- k / 2} {\tilde{p}}_{k} [λ (\cdot)]$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ , Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram); the algebra $A = R [{\tilde{p}}_{2}, {\tilde{p}}_{3}, \dots]$ ; and $Ω \in D_{0}$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ).

[F1]

For every partition $ρ$ with $r = ∣ ρ ∣$ and every $n \geq 0$ , $E_{P_{n}} [p_{ρ}^{#}] = n^{↓ r}$ for $ρ = (1^{r})$ and $E_{P_{n}} [p_{ρ}^{#}] = 0$ for $ρ \neq (1^{r})$ (Plancherel expectations of the shifted character observables).

[F2]

The scaling of profile moments is ${\tilde{p}}_{k} [\overset{ˉ}{λ}] = n^{- k / 2} {\tilde{p}}_{k} [λ (\cdot)]$ for $λ ⊢ n$ with $n \geq 1$ , and $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ for $ω \in D_{0}$ (Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$ , Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram).

[F3]

The family ${p_{ρ}^{#}}$ is a linear basis of $A$ , the weights $wt (p_{ρ}^{#}) = ∣ ρ ∣ + ℓ (ρ)$ define an algebra filtration of $A$ coinciding with the weight filtration generated by $wt ({\tilde{p}}_{k}) = k$ , and this filtration dominates the degree filtration $\deg_{1} (p_{ρ}^{#}) = ∣ ρ ∣ + m_{1} (ρ) \leq wt (p_{ρ}^{#})$ (The shifted character observables form a basis of $A$ , with the Kerov weight filtration).

[F4]

The functionals $L_{k} (p_{ρ}^{#}) = 1$ for $k$ even and $ρ = (1^{k / 2})$ , and $L_{k} (p_{ρ}^{#}) = 0$ otherwise, are multiplicative on weight-homogeneous elements, and $L_{2 m} ({\tilde{p}}_{2 m}) = (\binom{2 m}{m})$ and $L_{k} ({\tilde{p}}_{k}) = 0$ for odd $k$ (The profile-moment generators in the shifted-character basis).

[F5]

For the limit profile, ${\tilde{p}}_{2 m} [Ω] = (\binom{2 m}{m})$ for $m \geq 1$ and ${\tilde{p}}_{k} [Ω] = 0$ for odd $k$ (The profile moments of $Ω$ are central binomial coefficients).

[F6]

$Ω \in D_{0}$ with $σ_{Ω} = \frac{1}{2} (Ω - ∣ x ∣)$ supported in $[- 2, 2]$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ).

[F7]

Convergence in probability means for every $ε > 0$ , $P (∣ X_{n} - X ∣ > ε) \to 0$ (Convergence in probability), and for a random variable $Y$ with finite variance and mean $μ$ , $P (∣ Y - μ ∣ \geq ε) \leq Var (Y) / ε^{2}$ (Chebyshev's inequality for random variables).

Proof

technique · direct

1.1givenF2F3algebra

Monomial expansion: let $m = {\tilde{p}}_{j_{1}} \dots {\tilde{p}}_{j_{i}}$ be a monomial in the generators of $A$ of total weight $k = j_{1} + \dots + j_{i}$ , and fix $n \geq 1$ ; then $m$ lies in the weight filtration level $k$ of [F3], so its expansion $m = \sum_{ρ} m_{ρ} p_{ρ}^{#}$ in the basis of [F3] has $m_{ρ} = 0$ whenever $wt (p_{ρ}^{#}) > k$ ; and by [F2] the scaling identity applied to the $i$ factors gives $m (\overset{ˉ}{λ}) = n^{- k / 2} m (λ (\cdot)) = n^{- k / 2} \sum_{wt (ρ) \leq k} m_{ρ} p_{ρ}^{#} (λ)$ .

1.2givenF2F6algebra

Pointwise integral identity: for every $k \geq 0$ , every $λ ⊢ n$ and $ω \in D_{0}$ the definition of the profile moments in [F2] gives ${\tilde{p}}_{k + 2} [ω] = \frac{(k + 1) (k + 2)}{2} \int_{R} x^{k} (ω (x) - ∣ x ∣) d x$ ; subtracting the same identity for $ω = Ω$ , which lies in $D_{0}$ by [F6], yields the pointwise identity $\int_{R} (\overset{ˉ}{λ} (x) - Ω (x)) x^{k} d x = \frac{2}{(k + 1) (k + 2)} ({\tilde{p}}_{k + 2} [\overset{ˉ}{λ}] - {\tilde{p}}_{k + 2} [Ω])$ on $Y_{n}$ .

2.1givenF1step 1.1algebra

Limit of expectations: by step 1.1 and [F1], $E_{P_{n}} [m (\overset{ˉ}{λ})] = n^{- k / 2} \sum_{2 r \leq k} m_{(1^{r})} n^{↓ r}$ , and $n^{- k / 2} n^{↓ r} = n^{r - k / 2} \prod_{s = 0}^{r - 1} (1 - s / n)$ tends to $1$ when $2 r = k$ and to $0$ when $2 r < k$ ; hence $E_{P_{n}} [m (\overset{ˉ}{λ})]$ tends to $m_{(1^{k / 2})}$ if $k$ is even and to $0$ if $k$ is odd. Define the linear functional $L$ on $A$ by $L (m) : = m_{(1^{k / 2})}$ for a monomial of even weight $k$ and $L (m) : = 0$ for $k$ odd, extended linearly over the finitely many monomials of an element of $A$ ; then $E_{P_{n}} [f (\overset{ˉ}{λ})] \to L (f)$ for every $f \in A$ .

3.1givenF3F4step 2.1algebra

Multiplicativity: the functional $L$ of step 2.1 is the linear extension of the functionals $L_{k}$ of [F4], since for a monomial of weight $k$ the coefficient of $p_{(1^{k / 2})}^{#}$ is exactly the value of $L_{k}$ on its weight- $k$ component and lower-weight components contribute nothing; hence by the multiplicativity clause of [F4], applied to the weight-homogeneous components of two monomials, $L (m m^{'}) = L (m) L (m^{'})$ for all monomials, and by linearity $L$ is multiplicative on $A$ .

4.1givenF4F5step 2.1step 3.1algebra

Evaluation at the limit profile: by [F4] one has $L ({\tilde{p}}_{2 m}) = (\binom{2 m}{m})$ and $L ({\tilde{p}}_{k}) = 0$ for odd $k$ , while by [F5] the evaluation functional $f \mapsto f [Ω]$ has exactly the same values on the generators ${\tilde{p}}_{k}$ and is multiplicative with $1 [Ω] = 1$ ; since $A$ is generated by the ${\tilde{p}}_{k}$ , $L (f) = f [Ω]$ for every $f \in A$ , and step 2.1 gives $E_{P_{n}} [f (\overset{ˉ}{λ})] \to f [Ω]$ for every $f \in A$ .

5.1givenF7step 4.1algebra

Convergence of the moments in probability: applying step 4.1 to $f = {\tilde{p}}_{j}$ and to $f = {\tilde{p}}_{j}^{2}$ , and using that evaluation at $Ω$ is multiplicative, gives $E_{P_{n}} [{\tilde{p}}_{j} [\overset{ˉ}{λ}]] \to {\tilde{p}}_{j} [Ω]$ and $E_{P_{n}} [{\tilde{p}}_{j} [\overset{ˉ}{λ}]^{2}] \to {\tilde{p}}_{j} [Ω]^{2}$ , so the variances of the random variables ${\tilde{p}}_{j} [\overset{ˉ}{λ}]$ on the finite probability space $(Y_{n}, P_{n})$ tend to $0$ ; for every $ε > 0$ , the mean differs from ${\tilde{p}}_{j} [Ω]$ by less than $ε / 2$ for all sufficiently large $n$ . The event $∣ {\tilde{p}}_{j} [\overset{ˉ}{λ}] - {\tilde{p}}_{j} [Ω] ∣ \geq ε$ is then contained in the event of deviation at least $ε / 2$ from the current mean, so Chebyshev [F7] gives $P (∣ {\tilde{p}}_{j} [\overset{ˉ}{λ}] - {\tilde{p}}_{j} [Ω] ∣ \geq ε) \leq 4 Var ({\tilde{p}}_{j} [\overset{ˉ}{λ}]) / ε^{2} \to 0$ , that is, ${\tilde{p}}_{j} [\overset{ˉ}{λ}] \to {\tilde{p}}_{j} [Ω]$ in probability for every $j \geq 2$ .

6.1givenstep 1.2step 4.1step 5.1∎

Conclusion: for each fixed integer $k \geq 0$ , put $j = k + 2$ ; step 5.1 gives ${\tilde{p}}_{j} [\overset{ˉ}{λ}] \to {\tilde{p}}_{j} [Ω]$ in probability, and the identity of step 1.2 exhibits $\int_{R} (\overset{ˉ}{λ} - Ω) x^{k} d x$ as a fixed nonzero multiple of the difference, so $\int_{R} (\overset{ˉ}{λ} - Ω) x^{k} d x \to 0$ in probability as well; conversely the same identity transfers the latter convergence to the former. This proves the integral display and its equivalent moment form, and step 4.1 proves the assertion about every $f \in A$ . No choice principle is used: all steps are finite computations on the finite probability spaces $(Y_{n}, P_{n})$ .

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Kerov's central limit theorem for normalized cycle characters

Statement

Assume AC (The Axiom of Choice). For every fixed integer $N \geq 2$ , as $n \to \infty$ under the Plancherel measures $P_{n}$ , $(η_{k}^{(n)})_{2 \leq k \leq N} ⟹ N_{N - 1} (0, I_{N - 1}),$ that is, the $N - 1$ normalized cycle-character observables of Joint convergence in distribution and the normalized cycle-character observables converge jointly in distribution to independent standard Gaussians. Equivalently, $(\frac{p_{k}^{#}}{n^{k / 2}})_{2 \leq k \leq N} ⟹ (ζ_{2}, \dots, ζ_{N}),$ where the $ζ_{k}$ are independent centered Gaussians of variances $k$ . Convergence is the joint convergence of Joint convergence in distribution and the normalized cycle-character observables.

Facts & Assumptions

Given: AC; a fixed integer $N \geq 2$ ; the monic Hermite polynomials $H_{m}$ with $H_{0} = 1$ (The monic probabilists' Hermite polynomials); the normalized observables $η_{ρ}^{(n)} = p_{ρ}^{#} / (n^{∣ ρ ∣_{1} / 2} \prod_{k \geq 2} k^{m_{k} (ρ) / 2})$ with $∣ ρ ∣_{1} = ∣ ρ ∣ + m_{1} (ρ)$ and $η_{k}^{(n)} = p_{k}^{#} / (\sqrt{k} n^{k / 2})$ (Normalized shifted character observables $η_{ρ}$ , Joint convergence in distribution and the normalized cycle-character observables); and the probability spaces $(Y_{n}, P_{n})$ of The Plancherel measure on the partitions of $n$ .

[F1]

For every partition $ρ$ with $m_{1} (ρ) = 0$ and every $n \geq \max {1, ∣ ρ ∣}$ , $\prod_{k \geq 2} H_{m_{k} (ρ)} (η_{k}^{(n)}) = η_{ρ}^{(n)} + R_{ρ}^{(n)}$ with $∣ E_{P_{n}} [\prod_{k \geq 2} H_{m_{k} (ρ)} (η_{k}^{(n)})] - E_{P_{n}} [η_{ρ}^{(n)}] ∣ = O (n^{- 1 / 2})$ ; in particular the expectation of the Hermite product is $O (n^{- 1 / 2})$ when $ρ \neq \emptyset$ and equals $1$ when $ρ = \emptyset$ (Hermite leading terms for normalized shifted characters).

[F2]

For $Z \sim N (0, 1)$ : $E [H_{m} (Z)] = 0$ for every $m \geq 1$ and $E [H_{m} (Z) H_{n} (Z)] = m! δ_{m n}$ ; and for any $m_{1}, \dots, m_{N} \in N$ the mixed monomial $\prod_{k} x_{k}^{m_{k}}$ is a $Z$ -linear combination of products $\prod_{k} H_{m_{k}^{'}} (x_{k})$ with $m_{k}^{'} \leq m_{k}$ and $m_{k}^{'} \equiv m_{k} (m o d 2)$ , the coefficient of $\prod_{k} H_{m_{k}}$ being $1$ (Gaussian orthogonality and the monomial expansion of the Hermite polynomials).

[F3]

$E_{P_{n}} [η_{ρ}^{(n)}] = 0$ whenever $m_{1} (ρ) = 0$ and $ρ \neq \emptyset$ ; and $η_{k}^{(n)} = p_{k}^{#} / (\sqrt{k} n^{k / 2})$ for $2 \leq k \leq N$ (Normalized shifted character observables $η_{ρ}$ , Joint convergence in distribution and the normalized cycle-character observables).

[F4]

Joint convergence means weak convergence of the laws $μ_{n}$ on $R^{N - 1}$ ; the target law $N_{N - 1} (0, I_{N - 1})$ is the law of a vector with independent standard normal coordinates $ξ_{2}, \dots, ξ_{N}$ , whereas for $ζ_{k} : = \sqrt{k} ξ_{k}$ the law of $(ζ_{2}, \dots, ζ_{N})$ is $N_{N - 1} (0, diag (2, \dots, N))$ (Joint convergence in distribution and the normalized cycle-character observables, Multivariate normal law, including singular covariance).

[F5]

Under AC, if $R^{d}$ -valued random vectors have all mixed moments converging to those of a Borel probability $μ$ with finite moments that is determined by its mixed moments, then their laws converge weakly to $μ$ ; every multivariate Gaussian law is moment-determinate (The multivariate method of moments for a determinate limit).

[F6]

If $X_{2}, \dots, X_{N}$ are independent real random variables and $g_{k}$ Borel measurable with $g_{k} (X_{k})$ integrable, then $E [\prod_{k} g_{k} (X_{k})] = \prod_{k} E [g_{k} (X_{k})]$ (Expectations factor over finite products of independent random variables).

[F7]

For $Z \sim N (0, 1)$ one has $E [Z^{2 m}] = (2 m - 1)!!$ and $E ∣ Z ∣^{k} < \infty$ for every $k$ ; hence every polynomial in $Z$ is integrable (Gaussian even moments for Brownian increments, Cauchy-Schwarz for random variables, Standard normal and normal laws).

Proof

technique · direct

1.1givenF1F2F3F6F7algebra

Hermite-product moments: fix nonnegative integers $m_{2}, \dots, m_{N}$ and put $ρ : = (2^{m_{2}}, \dots, N^{m_{N}})$ , so $m_{1} (ρ) = 0$ and $ρ = \emptyset$ exactly when all $m_{k} = 0$ . If all $m_{k} = 0$ , then both $E_{P_{n}} [\prod_{k} H_{m_{k}} (η_{k}^{(n)})] = 1$ , using $H_{0} = 1$ , and $\prod_{k} E [H_{m_{k}} (ξ_{k})] = 1$ by [F2]; if some $m_{k} \geq 1$ then $ρ \neq \emptyset$ , and [F1] with [F3] gives $E_{P_{n}} [\prod_{k} H_{m_{k}} (η_{k}^{(n)})] = E_{P_{n}} [η_{ρ}^{(n)}] + O (n^{- 1 / 2}) = O (n^{- 1 / 2}) \to 0$ , while $\prod_{k} E [H_{m_{k}} (ξ_{k})] = 0$ by [F2] and [F6] because some factor has $m_{k} \geq 1$ and the remaining factors are integrable by [F7]. Hence $E_{P_{n}} [\prod_{k} H_{m_{k}} (η_{k}^{(n)})] \to \prod_{k} E [H_{m_{k}} (ξ_{k})]$ for every tuple $(m_{k})_{2 \leq k \leq N}$ .

2.1givenF2F6F7step 1.1algebra

Monomial moments: let $m_{2}, \dots, m_{N} \geq 0$ . By the expansion clause of [F2] the mixed monomial $\prod_{k} x_{k}^{m_{k}}$ equals a finite $Z$ -linear combination $\sum_{j} c_{j} \prod_{k} H_{j_{k}} (x_{k})$ ; evaluating at $x_{k} = η_{k}^{(n)}$ , taking expectations and using step 1.1 termwise for the finitely many tuples $(j_{k})$ gives $E_{P_{n}} [\prod_{k} (η_{k}^{(n)})^{m_{k}}] \to \sum_{j} c_{j} \prod_{k} E [H_{j_{k}} (ξ_{k})]$ ; evaluating the same expansion at $x_{k} = ξ_{k}$ and using [F6] and [F7] gives $E [\prod_{k} ξ_{k}^{m_{k}}] = \sum_{j} c_{j} \prod_{k} E [H_{j_{k}} (ξ_{k})]$ . Hence all mixed moments of $(η_{2}^{(n)}, \dots, η_{N}^{(n)})$ converge to the corresponding mixed moments of the standard Gaussian vector $(ξ_{2}, \dots, ξ_{N})$ .

3.1givenF4F5step 2.1

Convergence in distribution: by step 2.1 hypothesis (i) of [F5] holds for the vectors $X_{n} : = (η_{2}^{(n)}, \dots, η_{N}^{(n)})$ , $d : = N - 1$ , and the target $μ : = N_{N - 1} (0, I_{N - 1})$ , which by [F4] is the law of $(ξ_{2}, \dots, ξ_{N})$ ; hypothesis (ii) is the Gaussian determinacy clause of [F5]; hence $η^{(n)} ⟹ N_{N - 1} (0, I_{N - 1})$ .

3.2givenF3F4F5F6step 2.1algebra

Equivalent unnormalized form: by [F3], $p_{k}^{#} / n^{k / 2} = \sqrt{k} η_{k}^{(n)}$ for each $k$ , so for every tuple $(m_{k})$ the moment $E_{P_{n}} [\prod_{k} (p_{k}^{#} / n^{k / 2})^{m_{k}}]$ equals $\prod_{k} k^{m_{k} / 2} E_{P_{n}} [\prod_{k} (η_{k}^{(n)})^{m_{k}}]$ and converges by step 2.1 to $\prod_{k} k^{m_{k} / 2} E [\prod_{k} ξ_{k}^{m_{k}}] = E [\prod_{k} ζ_{k}^{m_{k}}]$ , where the last equality uses $ζ_{k} = \sqrt{k} ξ_{k}$ and the factorization [F6]; the law of $(ζ_{2}, \dots, ζ_{N})$ is the multivariate Gaussian law $N_{N - 1} (0, Σ)$ with $Σ = diag (2, \dots, N)$ of [F4], which is moment-determinate by [F5], so a second application of [F5] gives $(p_{k}^{#} / n^{k / 2})_{2 \leq k \leq N} ⟹ (ζ_{2}, \dots, ζ_{N})$ .

4.1givenF4F5step 3.1step 3.2∎

Conclusion: step 3.1 proves the normalized convergence and step 3.2 its stated equivalent form, both for every fixed $N \geq 2$ . AC is used exactly through [F5] (Prokhorov, Skorokhod and the Gaussian determinacy clause) and the target law of [F4].

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Plancherel Young diagrams converge to the limit shape

Statement

Let $λ$ range over $Y_{n}$ under the Plancherel measure $P_{n}$ and let $\overset{ˉ}{λ}$ be the $\sqrt{n}$ -scaled Russian profile of Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram. Then $\sup_{x \in R} ∣ \overset{ˉ}{λ} (x) - Ω (x) ∣ ⟶ 0 in probability as n \to \infty,$ where $Ω$ is the limit profile of The Logan-Shepp-Vershik-Kerov limit profile $Ω$ .

Facts & Assumptions

Given: the probability space $(Y_{n}, P_{n})$ (The Plancherel measure on the partitions of $n$ , The Plancherel weights sum to one); the profiles $\overset{ˉ}{λ}$ and $Ω$ and their $σ$ -functions $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ (Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram, The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ); a constant $C > e$ .

[F1]

For $C > e$ there is $n_{0}$ such that for all $n \geq n_{0}$ the event $E_{n} : = {λ_{1} \leq C \sqrt{n} and λ_{1}^{'} \leq C \sqrt{n}}$ satisfies $P (E_{n}) \geq 1 - 2 (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1} \to 1$ , and on $E_{n}$ the function $x \mapsto \overset{ˉ}{λ} (x) - ∣ x ∣$ is supported in $[- C, C]$ (The RSK union bound localizes Plancherel profiles).

[F2]

Fix $I = [a, b]$ and let $Σ_{I}$ be the set of real functions supported in $I$ with $∣ σ (x) - σ (y) ∣ \leq ∣ x - y ∣$ . For every $ε > 0$ there are $K \in N$ and $δ > 0$ such that every $σ \in Σ_{I}$ with $∣ \int_{R} σ (x) x^{k} d x ∣ \leq δ$ for $k = 0, 1, \dots, K$ satisfies $\sup_{x \in R} ∣ σ (x) ∣ \leq ε$ (Finitely many polynomial moments control the uniform distance on bounded Lipschitz profiles).

[F3]

Every $ω \in D_{0}$ has $σ_{ω} = \frac{1}{2} (ω - ∣ x ∣)$ $1$ -Lipschitz; the support of $σ_{λ}$ is contained in $[- λ_{1}^{'}, λ_{1}]$ , and $σ_{\overset{ˉ}{λ}} (x) = n^{- 1 / 2} σ_{λ} (n^{1 / 2} x)$ , so $σ_{\overset{ˉ}{λ}}$ is supported in $[- λ_{1}^{'} / \sqrt{n}, λ_{1} / \sqrt{n}]$ (Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram); $Ω \in D_{0}$ with $σ_{Ω}$ supported in $[- 2, 2]$ (The Logan-Shepp-Vershik-Kerov limit profile $Ω$ ).

[F4]

For every integer $k \geq 0$ , $\int_{R} (\overset{ˉ}{λ} (x) - Ω (x)) x^{k} d x \to 0$ in probability as $n \to \infty$ (Scaled Plancherel profile moments converge in probability).

[F5]

Probability is subadditive, $P (⋃_{j} A_{j}) \leq \sum_{j} P (A_{j})$ for finitely many events (Basic identities for a probability measure); convergence in probability means for every $ε > 0$ , $P (∣ X_{n} - X ∣ > ε) \to 0$ (Convergence in probability).

Proof

technique · direct

1.1givenF1F3algebra

The test profile: put $I : = [- C, C]$ and $g : = \frac{1}{2} (σ_{\overset{ˉ}{λ}} - σ_{Ω}) = \frac{1}{4} (\overset{ˉ}{λ} - Ω)$ . On $E_{n}$ the function $σ_{\overset{ˉ}{λ}}$ is supported in $[- λ_{1}^{'} / \sqrt{n}, λ_{1} / \sqrt{n}] \subseteq I$ by [F1] and [F3], and $σ_{Ω}$ is supported in $[- 2, 2] \subseteq I$ because $C > e > 2$ ; hence $g$ is supported in $I$ . Moreover by [F3] both $σ_{\overset{ˉ}{λ}}$ and $σ_{Ω}$ are $1$ -Lipschitz, so $∣ g (x) - g (y) ∣ \leq \frac{1}{2} (∣ σ_{\overset{ˉ}{λ}} (x) - σ_{\overset{ˉ}{λ}} (y) ∣ + ∣ σ_{Ω} (x) - σ_{Ω} (y) ∣) \leq ∣ x - y ∣$ ; thus $g \in Σ_{I}$ on $E_{n}$ .

2.1givenF2step 1.1algebra

Deterministic containment: fix $ε > 0$ and apply [F2] with tolerance $ε / 4$ to obtain $K \in N$ and $δ > 0$ such that $σ \in Σ_{I}$ and $∣ \int σ x^{k} ∣ \leq δ$ for $k = 0, \dots, K$ imply $\sup ∣ σ ∣ \leq ε / 4$ . On $E_{n}$ , if $\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ > ε$ then $\sup ∣ g ∣ = \frac{1}{4} \sup ∣ \overset{ˉ}{λ} - Ω ∣ > ε / 4$ , so by the contrapositive of the lemma there is $k \in {0, \dots, K}$ with $∣ \int g x^{k} ∣ > δ$ , that is, $∣ \int (\overset{ˉ}{λ} - Ω) x^{k} ∣ > 4 δ$ because $g = \frac{1}{4} (\overset{ˉ}{λ} - Ω)$ ; hence on $E_{n}$ the event ${\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ > ε}$ is contained in $⋃_{k = 0}^{K} {∣ \int_{R} (\overset{ˉ}{λ} - Ω) x^{k} d x ∣ > 4 δ}$ , and consequently ${\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ > ε} \subseteq E_{n}^{c} \cup ⋃_{k = 0}^{K} {∣ \int_{R} (\overset{ˉ}{λ} - Ω) x^{k} d x ∣ > 4 δ}$ .

3.1givenF1F4F5step 2.1algebra∎

Probability bound: by [F5] and step 2.1, $P (\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ > ε) \leq P (E_{n}^{c}) + \sum_{k = 0}^{K} P (∣ \int (\overset{ˉ}{λ} - Ω) x^{k} ∣ > 4 δ)$ . Here $P (E_{n}^{c}) \leq 2 (e^{2} / C^{2})^{⌊ C \sqrt{n} ⌋ - 1} \to 0$ by [F1], and each of the finitely many terms tends to $0$ by [F4] and the definition of convergence in probability in [F5]. Hence $P (\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ > ε) \to 0$ ; since $ε > 0$ was arbitrary, $\sup_{x} ∣ \overset{ˉ}{λ} - Ω ∣ \to 0$ in probability.

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The RSK and longest-increasing-subsequence consequences remain owned by the hook-length/RSK page

Remark

This page consumes the distribution of the Robinson-Schensted shape of a uniform permutation (The RSK shape of a uniform random permutation has the Plancherel law) and Schensted's longest increasing and decreasing subsequence theorem only to localize Plancherel profiles (The RSK union bound localizes Plancherel profiles). It does not re-mint RSK, the LIS/LDS identities, the Baik-Deift-Johansson theorem, the Tracy-Widom distribution, determinantal point processes or edge statistics of Plancherel measure: those belong to the hook-length/RSK page and to separate analytic-probability suppliers, and the limit-shape theorem proved here (Plancherel Young diagrams converge to the limit shape) is the qualitative law of large numbers, not a fluctuation or edge result. In particular no sharp constant, rate or fluctuation-distribution statement is asserted: the localization lemma gives only the bound $λ_{1}, λ_{1}^{'} \leq C \sqrt{n}$ with probability tending to one, for each fixed $C > e$ , and the limit-shape theorem gives only convergence in probability of the scaled profile to $Ω$ .

5 · Examples, counterexamples and false statements

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Plancherel Measure and Asymptotic Young Diagrams

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The monic probabilists' Hermite polynomials

Definition

The Plancherel measure on the partitions of nn

Definition

Finitely many polynomial moments control the uniform distance on bounded Lipschitz profiles

Statement

Facts & Assumptions

Proof

The standard Gaussian law is determined by its moments

Statement

Facts & Assumptions

Proof

Continual diagrams, Russian profiles, and the nn​-scaling of a Young diagram

Definition

Gaussian orthogonality and the monomial expansion of the Hermite polynomials

Statement

Facts & Assumptions

Proof

The Plancherel weights sum to one

Statement

Facts & Assumptions

Proof

The multivariate method of moments for a determinate limit

Statement

Facts & Assumptions

Proof

The RSK shape of a uniform random permutation has the Plancherel law

Statement

Facts & Assumptions

Proof

The Logan-Shepp-Vershik-Kerov limit profile ΩΩ

Definition

Shifted character observables pρ#pρ#​ and profile moments p~kp~​k​

Definition

The RSK union bound localizes Plancherel profiles

Statement

Facts & Assumptions

Proof

Joint convergence in distribution and the normalized cycle-character observables

Definition

The profile moments of ΩΩ are central binomial coefficients

Statement

Facts & Assumptions

Proof

Plancherel expectations of the shifted character observables

Statement

Facts & Assumptions

Proof

The shifted character observables form a basis of AA, with the Kerov weight filtration

Statement

Facts & Assumptions

Proof

Normalized shifted character observables ηρηρ​

Definition

The profile-moment generators in the shifted-character basis

Statement

Facts & Assumptions

Proof

Shifted character products: exact for p1#p1#​ and leading terms for pk#pk#​

Statement

Facts & Assumptions

Proof

Hermite leading terms for normalized shifted characters

Statement

Facts & Assumptions

Proof

Scaled Plancherel profile moments converge in probability

Statement

Facts & Assumptions

Proof

Kerov's central limit theorem for normalized cycle characters

Statement

Facts & Assumptions

Proof

Plancherel Young diagrams converge to the limit shape

The Plancherel measure on the partitions of $n$

Continual diagrams, Russian profiles, and the $\sqrt{n}$ -scaling of a Young diagram

The Logan-Shepp-Vershik-Kerov limit profile $Ω$

Shifted character observables $p_{ρ}^{#}$ and profile moments ${\tilde{p}}_{k}$

The profile moments of $Ω$ are central binomial coefficients

The shifted character observables form a basis of $A$ , with the Kerov weight filtration

Normalized shifted character observables $η_{ρ}$

Shifted character products: exact for $p_{1}^{#}$ and leading terms for $p_{k}^{#}$