Expectations of hook products on large partitions and the chi-square distribution

Abstract

Given uniform probability on words of length M = Np + k, from an alphabet of size p, consider the probability that a word (i) contains a subsequence of letters p, p − 1, …, 1 in that order and (ii) that the maximal length of the disjoint union of p − 1 increasing subsequences of the word is ⩽ M − N. A generating function for this probability has the form of an integral over the Grassmannian of p-planes in ℂ^n. The present paper shows that the asymptotics of this probability, when N → ∞, is related to the kth moment of the χ^2-distribution of parameter 2p^2. This is related to the behavior of the integral over the Grassmannian Gr(p, ℂ^n) of p-planes in ℂ^n, when the dimension of the ambient space ℂ^n becomes very large. A dierent scaling limit for the Poissonized probability is related to a new matrix integral, itself a solution of the Painlevé IV equation. This is part of a more general set-up related to the Painlevé V equation.

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