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Aspects of Toeplitz determinants

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arxiv 1007.1128 v3 pith:JE2HCXKA submitted 2010-07-07 math-ph math.CAmath.FAmath.MP

I. Krasovsky This is my paper · ORCID
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keywords determinantstoeplitzhankelrelatedasymptoticscertainfredholmrandom
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We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

    math-ph 2019-09 accept novelty 8.0 of 10

    A 4x4 Riemann-Hilbert formalism gives large-n asymptotics for Toeplitz+Hankel determinants with independent symbols, conditional on a non-degeneracy bound and verified for an explicit family.

  2. Emptiness formation probability and Painlev\'e V equation in the XY spin chain

    cond-mat.stat-mech 2019-09 conditional novelty 6.0 of 10

    The emptiness formation probability of the XY chain, in the double-scaling limit near its critical lines, is governed by a Painleve V tau function; the result is exact at the Ising point and numerically supported elsewhere.

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