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

Adler, Mark and Borodin, Alexei and van Moerbeke, Pierre (2007) Expectations of hook products on large partitions and the chi-square distribution. Forum Mathematicum, 19 (1). pp. 159-186. ISSN 0933-7741. doi:10.1515/FORUM.2007.008. https://resolver.caltech.edu/CaltechAUTHORS:20180808-094303177

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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 ⩽ 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1515/FORUM.2007.008DOIArticle
Additional Information:© 2007 de Gruyter. Received: 2006-01-03; Published Online: 2007-02-21; Published in Print: 2007-01-29. The support of a National Science Foundation grant a DMS-01-00782 is gratefully acknowledged. The support of a National Science Foundation grant a DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. This work was done while PvM was a member of the Clay Mathematics Institute, One Bow Street, Cambridge, MA 02138, USA. MA and PvM would like to thank Persi Diaconis for interesting conversations regarding section 4.
Funders:
Funding AgencyGrant Number
NSFDMS-01-00782
North Atlantic Treaty Organization (NATO)UNSPECIFIED
Fonds de la Recherche Scientifique (FNRS)UNSPECIFIED
Francqui FoundationUNSPECIFIED
Issue or Number:1
Classification Code:2000 Mathematics Subject Classification: 60C55, 60F05, 05A05, 05A15, 34M55
DOI:10.1515/FORUM.2007.008
Record Number:CaltechAUTHORS:20180808-094303177
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180808-094303177
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88650
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:08 Aug 2018 17:24
Last Modified:16 Nov 2021 00:28

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