1 November 2011 Emergence of a singularity for Toeplitz determinants and Painlevé V
T. Claeys, A. Its, I. Krasovsky
Duke Math. J. 160(2): 207-262 (1 November 2011). DOI: 10.1215/00127094-1444207


We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular so that the determinants obey Szegő’s strong limit theorem. If t=0, the symbol possesses a Fisher-Hartwig singularity. Letting t0 we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.


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T. Claeys. A. Its. I. Krasovsky. "Emergence of a singularity for Toeplitz determinants and Painlevé V." Duke Math. J. 160 (2) 207 - 262, 1 November 2011. https://doi.org/10.1215/00127094-1444207


Published: 1 November 2011
First available in Project Euclid: 27 October 2011

zbMATH: 1298.47039
MathSciNet: MR2852117
Digital Object Identifier: 10.1215/00127094-1444207

Rights: Copyright © 2011 Duke University Press


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Vol.160 • No. 2 • 1 November 2011
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