The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix

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 $m_n^{\prime }$ 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 $m_n^{\prime }$ can be written as a sum over planar diagrams with two terms, ${\sum }_{\pi }{c}^{\prime }(\#(\pi )-1)c^{\#(\pi )-1}$, 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.

arXiv comment36 pages, updated the references, added some comments, main results unchanged

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