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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].

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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)
Piotr Sniady, Gaussian fluctuations of characters of symmetric groups and of Young diagrams, arXiv:math/0501112 (standard reference, not scraped)