Expectations of hook products on large partitions
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.
Cite
@article{arxiv.math/0409554,
title = {Expectations of hook products on large partitions},
author = {Mark Adler and Alexei Borodin and Pierre van Moerbeke},
journal= {arXiv preprint arXiv:math/0409554},
year = {2016}
}
Comments
31 pages, 1 figure