We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter . For positive, the symbols are regular so that the determinants obey Szegő’s strong limit theorem. If , the symbol possesses a Fisher-Hartwig singularity. Letting 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.
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