Expectations of hook products on large partitions and the chi-square distribution
-
Mark Adler
,
Alexei Borodin
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 ℂn. The present paper shows that the asymptotics of this probability, when N → ∞, is related to the kth moment of the χ2-distribution of parameter 2p2. This is related to the behavior of the integral over the Grassmannian Gr(p, ℂn) of p-planes in ℂn, when the dimension of the ambient space ℂn 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.
© Walter de Gruyter
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Artikel in diesem Heft
- Solving Abstract Cauchy Problems with closable operators in reflexive spaces via resolvent-free approximation
- Chain transitive sets for flows on flag bundles
- A Burgess-like subconvex bound for twisted L-functions
- On recurrence in zero dimensional flows
- Solutions of nonlinear elliptic equations in unbounded Lipschitz domains
- π∗(L2T(1)/(v1)) and its applications in computing π∗(L2T(1)) at the prime two
- A proof of the Livingston conjecture
- Expectations of hook products on large partitions and the chi-square distribution
