Emergence of a singularity for Toeplitz determinants and Painlevé V
(2011) Duke Mathematical Journal — Vol. 160, n° 2, p. 207-262 (2011)
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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 Szegő’s strong limit theorem. If t=0, the symbol possesses a Fisher-Hartwig singularity. Letting t→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 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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Claeys, T., Its, A., & Krasovsky, I. (2011). Emergence of a singularity for Toeplitz determinants and Painlevé V. Duke Mathematical Journal, 160(2), 207-262. https://doi.org/10.1215/00127094-1444207 (Original work published 2011)