Expectations of hook products on large partitions and the chi-square distribution
(2007) Forum mathematicum — Vol. 19, n° 1, p. 159-186 (2007)
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- 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 -> infinity, is related to the k(th) 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 Painleve IV equation. This is part of a more general set-up related to the Painleve V equation.
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Adler, M., Borodin, A., & Van Moerbeke, P. (2007). Expectations of hook products on large partitions and the chi-square distribution. Forum mathematicum, 19(1), 159-186. https://doi.org/10.1515/FORUM.2007.008 (Original work published 2007)