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Interpretations of some parameter dependent generalizations of classical matrix ensembles

Forrester, Peter J. and Rains, Eric M. (2005) Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probability Theory and Related Fields, 131 (1). pp. 1-61. ISSN 0178-8051. doi:10.1007/s00440-004-0375-6.

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Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Anderson’s and Dixon’s work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.

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Additional Information:© 2004 Springer-Verlag Berlin Heidelberg. Received: 26 November 2002; Revised: 07 May 2004; First Online: 20 August 2004. Supported by the Australian Research Council.
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Australian Research CouncilUNSPECIFIED
Issue or Number:1
Record Number:CaltechAUTHORS:20171003-160005064
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Official Citation:Forrester, P. & Rains, E. Probab. Theory Relat. Fields (2005) 131: 1.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82017
Deposited By: Tony Diaz
Deposited On:04 Oct 2017 14:45
Last Modified:15 Nov 2021 19:47

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