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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.

Depends on

Used by

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

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Vladimir Ivanov and Grigori Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, arXiv:math/0304010; survey-paper version in Symmetric Functions 2001, NATO Science Series II 74 (2002), 93-151 (standard reference, not scraped)