Emergence of a singularity for Toeplitz determinants and Painlevé V

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 $t\to 0$ 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 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.