Emergence of a singularity for Toeplitz determinants and Painleve V

TL;DR

This paper studies the asymptotic behavior of Toeplitz determinants with symbols transitioning from regular to singular, revealing a phase transition described by Painleve V and connecting to the Ising model's spin correlations.

Contribution

It provides a detailed analysis of the emergence of Fisher-Hartwig singularities in Toeplitz determinants and links this transition to Painleve V transcendents, extending classical results.

Findings

Asymptotic expansions for Toeplitz determinants with parameter-dependent symbols.

Identification of a phase transition described by Painleve V.

Connection to the behavior of 2-spin correlations in the Ising model.

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.