Emergence of a singularity for Toeplitz determinants and Painleve V

Details Emergence of a singularity for Toeplitz determinants and Painleve V Emergence of a singularity for Toeplitz determinants and Painleve V

2010-04-21

arxiv.org/abs/1004.3696v2

Duke Math. J. 160 (2011), 207-262

Physics

Mathematical Physics

46 pages

Scientific paper

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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