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Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter

Forrester, Peter J. and Rains, Eric M. (2004) Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter. Probability Theory and Related Fields, 130 (4). pp. 518-576. ISSN 0178-8051. doi:10.1007/s00440-004-0374-7.

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A superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original, while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue. In the cases of the classical matrix ensembles with orthogonal symmetry, it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry. The basic identities expressing these facts can be extended to include a parameter, which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles. The parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble, while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues. By the construction of new families of biorthogonal and skew orthogonal polynomials, we are able to compute the corresponding correlation functions, both in the finite system and in various scaled limits. Specializing back to the cases of orthogonal and symplectic symmetry, we find that our results imply different functional forms to those known previously.

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Additional Information:© 2004 Springer-Verlag Berlin Heidelberg. Received: 26 November 2002; Revised version: 23 March 2004; Published online: 5 July 2004. We thank J. Baik for encouraging us to take up the problem of calculating the correlation functions for (1.4), and the referee for a careful reading. The work of PJF was supported by the Australian Research Council.
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Australian Research CouncilUNSPECIFIED
Issue or Number:4
Record Number:CaltechAUTHORS:20171003-102014270
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Official Citation:Forrester, P. & Rains, E. Probab. Theory Relat. Fields (2004) 130: 518.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81989
Deposited By: Tony Diaz
Deposited On:03 Oct 2017 20:52
Last Modified:15 Nov 2021 19:47

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