The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix

TL;DR

This paper studies the detailed asymptotic behavior of eigenvalues of real Wishart matrices, revealing new combinatorial formulas for infinitesimal moments and connections to higher order free probability.

Contribution

It introduces a novel combinatorial representation for the infinitesimal moments of real Wishart matrices, linking them to planar diagrams and non-crossing annular permutations.

Findings

Derived a sum-over-diagrams formula for infinitesimal moments.

Connected the second term to higher order freeness recursion.

Provided explicit combinatorial structures for asymptotic eigenvalue fluctuations.

Abstract

Let $X_N$ be a $N \times N$ real Wishart random matrix with aspect ratio $M/N$. The limit eigenvalue distribution of $X_N$ is the Marchenko-Pastur law with parameter $c = \lim_N M/N$. The limit moments $\{m_n\}_n$ are given by $m_n = \sum_{\pi} c^{\#(\pi)}$ where the sum runs over $NC(n)$. Let $m_n'$ be the limit of $N( \mathrm{E}(\mathrm {tr}(X_N^n)) - m_n)$. These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that $m'_n$ can be written as a sum over planar diagrams with two terms, $\sum_{\pi} c'(\#(\pi) -1) c^{\#(\pi)-1}$, and $\sum_{\pi \in S_{NC}^\delta(n,-n)} c^{\#(\pi)/2}$, where $S_{NC}^\delta(n,-n)$ is a set of non-crossing annular permutations satisfying a symmetry condition. Moreover we present a recursion formula for the second term which is related to one for higher order freeness.