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On an Identity due to Bump and Diaconis, and Tracy and Widom

Published online by Cambridge University Press:  20 November 2018

Paul-Olivier Dehaye
Paul-Olivier Dehaye*
Affiliation:
Merton College, University of Oxford, United Kingdome-mail: paul-olivier.dehaye@math.ethz.ch
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Abstract

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A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

Keywords

47B3505E0520G05Toeplitz matricesJacobi–Trudi identitySzegö limit theoremHeine identityWiener–Hopf factorization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 255 - 269
Copyright
Copyright © Canadian Mathematical Society 2011

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