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Expectations of hook products on large partitions and the chi-square distribution.
- Source :
-
Forum Mathematicum . 2007, Vol. 19 Issue 1, p159-186. 28p. 1 Diagram. - Publication Year :
- 2007
-
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 χ2-distribution of parameter 2 p2. 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09337741
- Volume :
- 19
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Forum Mathematicum
- Publication Type :
- Academic Journal
- Accession number :
- 24099634
- Full Text :
- https://doi.org/10.1515/FORUM.2007.008