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Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Borodin, Alexei and Forrester, Peter J. (2003) Increasing subsequences and the hard-to-soft edge transition in matrix ensembles. Journal of Physics A: Mathematical and General, 36 (12). pp. 2963-2981. ISSN 0305-4470. http://resolver.caltech.edu/CaltechAUTHORS:BORjpa03

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Abstract

Our interest is in the cumulative probabilities Pr(L(t) ≤ l)for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik–Deift–Johansson and Baik–Rains.


Item Type:Article
Additional Information:Copyright © Institute of Physics and IOP Publishing Limited 2003 Received 28 August 2002, in final form 17 December 2002, Published 12 March 2003, Print publication: Issue 12 (28 March 2003) This research was partially conducted during the period AB served as a Clay Mathematics Institute Long-Term Prize Fellow, and was supported in part by the NSF grant DMS-9729992. The work of PJF was supported by the Australian Research Council. Thanks are due to Eric Rains for pointing out the significance of the zonal polynomial identities (2.19)–(2.21), and to Percy Deift for facilitating this collaboration by supporting a visit of PJF to the University of Pennsylvania during April 2001. Helpful comments of a referee are also acknowledged.
Record Number:CaltechAUTHORS:BORjpa03
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:BORjpa03
Alternative URL:http://dx.doi.org/10.1088/0305-4470/36/12/307
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2615
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:12 Apr 2006
Last Modified:26 Dec 2012 08:50

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