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Expectations of hook products on large partitions

Mark Adler·Alexei Borodin·Pierre van Moerbeke
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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 \leq M-N . A generating function for this probability has the form of an integral over the Grassmannian of p-planes in complex C^n. The present paper shows that the asymptotics of this probability, when N tends to infinity, is related to the kth moment of the chi^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 different scaling limit for the Poissonized probability is related to a new matrix integral, itself a solution of the Painlev\'e IV equation. This is part of a more general set-up related to the Painlev\'e V equation.

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Mark Adler, Alexei Borodin, Pierre van Moerbeke. 2004-09-28. Expectations of hook products on large partitions. https://arxiv.org/abs/math/0409554

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