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τ-function of discrete isomonodromy transformations and probability

Part of: Curves Difference equations Difference and functional equations

Published online by Cambridge University Press:  01 May 2009

D. Arinkin  and
A. Borodin
D. Arinkin
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA (email: arinkin@email.unc.edu)
A. Borodin
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA, USA (email: borodin@caltech.edu)
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Abstract

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We introduce the τ-function of a difference rational connection (d-connection) and its isomonodromy transformations. We show that in a continuous limit ourτ-function agrees with the Jimbo–Miwa–Ueno τ-function. We compute the τ-function for the isomonodromy transformations leading to difference Painlevé V and difference Painlevé VI equations. We prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the τ-function for an associated d-connection.

Keywords

discrete Painlevé equationsτ-functions

MSC classification

Secondary: 39A10: Difference equations, additive 14H60: Vector bundles on curves and their moduli
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 747 - 772
Copyright
Copyright © Foundation Compositio Mathematica 2009

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