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arXiv · 1004.3696

Emergence of a singularity for Toeplitz determinants and Painleve V

T. Claeys·A. Its·I. Krasovsky
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Abstract

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 Szeg\H{o}'s strong limit theorem. If $t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ 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 Painlev\'e 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. 2010-04-21. Emergence of a singularity for Toeplitz determinants and Painleve V. https://doi.org/10.1215/00127094-1444207

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