The asymptotic infinitesimal distribution of a real Wishart random matrix
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Abstract
Let XN be a N × N real Wishart random matrix with aspect ratio M/N. The limit eigenvalue distribution of XN is the Marchenko-Pastur law with parameter c = limNM/N. The limit moments {mn}n are given by mn = ∑πc#(π) where the sum runs over NC(n). Let mn′ be the limit of N(E(tr(XNn))−mn). These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that mn′ can be written as a sum over planar diagrams with two terms, ∑πc′(#(π) − 1)c#(π)−1, and ∑π∈SNCδ(n,−n)c#(π)/2, where SNCδ(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.
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