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A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

Forrester, Peter J. and Rains, Eric M. (2012) A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals. Communications in Mathematical Physics, 309 (3). pp. 771-792. ISSN 0010-3616. doi:10.1007/s00220-011-1305-y.

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We characterize averages of ∏^N_(l=1)│x−t_l│^(ɑ−1) with respect to the Selberg density, further constrained so that t_l є [0,x](l=1,...,q) and t_l є [x,1](l=q^+1,...,N), in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the t_j(j=1,...,N) . In the case q = 0 and α < 1 we compute the explicit leading order term in the x → 0 asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case q = 0 and ɑ є Z^+, and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.

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Additional Information:© 2011 Springer-Verlag. Received: 26 January 2011. Accepted: 15 February 2011. Published online: 18 August 2011. Communicated by S. Zelditch. The contribution to the preparation of this paper by Wendy Baratta and James Saunderson is acknowledged. This work was supported by the Australian Research Council.
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Australian Research CouncilUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20120314-084236569
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29716
Deposited By: Ruth Sustaita
Deposited On:14 Mar 2012 16:21
Last Modified:09 Nov 2021 19:28

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