Plancherel测度的Jack变形与无迹高斯随机矩阵
Jack deformations of Plancherel measures and traceless Gaussian random matrices
摘要：
研究了长度不大于固定数$d$的$n$的随机划分$λ=(λ_1,λ_2,...,λ_d)$.假设随机划分$λ$是按Jack测度分布的，Jack测度是Plancherel测度的一种变形，具有正参数$α>0$ .证明了对所有的$α>0$，在极限为$n \to \infty$的情况下，标度$λ_1,..., λ_d$的联合分布收敛于无迹高斯$β$ -系综与$β=2/α$的某些随机变量的联合分布.我们还给出了一个简短的证明Regev的渐近定理的总和$β$ -权力的$f^λ$，数量的标准tableaux的形状$λ$。
英文摘要：
We study random partitions $λ=(λ_1,λ_2,...,λ_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $λ$ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter $α>0$. We prove that for all $α>0$, in the limit as $n \to \infty$, the joint distribution of scaled $λ_1,..., λ_d$ converges to the joint distribution of some random variables from a traceless Gaussian $β$-ensemble with $β=2/α$. We also give a short proof of Regev's asymptotic theorem for the sum of $β$-powers of $f^λ$, the number of standard tableaux of shape $λ$.
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