圆β系综迹的矩
Moments of traces of circular beta-ensembles
摘要：
令$θ_1,\ldots,θ_n$是来自具有概率密度函数$\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{iθ_j}-e^{iθ_k}|^β$的Dyson循环$β$系综的随机变量。对于每个$n\geq2$和$β>0$，我们得到了关于$\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]$的一些不等式，其中$Z_n=(e^{iθ_1},\ldots,e^{iθ_n})$和$p_μ$是$μ$的幂和对称函数.当$β=2$时，我们的不等式恢复了Diaconis和Evans关于Haar不变酉矩阵的一个恒等式。此外，我们有以下：$ \lim_{n\to\infty}\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]= δ_{μν}(\frac{2}β)^{l(μ)}z_μ$用于任何$β>0$和分区$μ,ν$ ; $\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n$用于任何$β>0$和$n\geq2$，其中$l(μ)$是$μ$的长度，$z_μ$在$μ$上是显式的。这些结果适用于三个重要的合奏：COE（$β=1$），CUE（$β=2$）和CSE（$β=4$）。我们进一步研究了$β=1,4$的$\mathbb{E}[|p_m(Z_n)|^2]$的非渐近行为。当（i）$g(z)$是多项式且$β>0$是任意的，或（ii）$g(z)$有Fourier展开且$β=1,4$时，得到了$\sum_{j=1}^ng(e^{iθ_j})$的中心极限定理.主要工具是Jack功能。
英文摘要：
Let $θ_1,\ldots,θ_n$ be random variables from Dyson's circular $β$-ensemble with probability density function $\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{iθ_j}-e^{iθ_k}|^β$. For each $n\geq2$ and $β>0$, we obtain some inequalities on $\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]$, where $Z_n=(e^{iθ_1},\ldots,e^{iθ_n})$ and $p_μ$ is the power-sum symmetric function for partition $μ$. When $β=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $ \lim_{n\to\infty}\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]= δ_{μν}(\frac{2}β)^{l(μ)}z_μ$ for any $β>0$ and partitions $μ,ν$; $\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n$ for any $β>0$ and $n\geq2$, where $l(μ)$ is the length of $μ$ and $z_μ$ is explicit on $μ$. These results apply to the three important ensembles: COE ($β=1$), CUE ($β=2$) and CSE ($β=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_m(Z_n)|^2]$ for $β=1,4$. The central limit theorems of $\sum_{j=1}^ng(e^{iθ_j})$ are obtained when (i) $g(z)$ is a polynomial and $β>0$ is arbitrary, or (ii) $g(z)$ has a Fourier expansion and $β=1,4$. The main tool is the Jack function.
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