Natural Boundary for a Sum Involving Toeplitz Determinants

TL;DR

This paper proves that the diagonal susceptibility in the 2D Ising model, expressed via Toeplitz determinants, has the unit circle as a natural boundary, extending previous results to more general symbol deformations.

Contribution

It extends the proof of the natural boundary property from specific Toeplitz symbol deformations to broader classes including Fisher-Hartwig symbols.

Findings

The unit circle remains a natural boundary for a wider class of Toeplitz determinants.

The result applies to deformations of symbols with a single singularity on the unit circle.

The analysis covers almost general Fisher-Hartwig symbols.

Abstract

In the theory of the two-dimensional Ising model, the diagonal susceptibility is equal to a sum involving Toeplitz determinants. In terms of a parameter k the diagonal susceptibility is analytic inside the unit circle, and the authors proved the conjecture that this function has the unit circle as a natural boundary. The symbol of the Toepltiz determinants was a k-deformation of one with a single singularity on the unit circle. Here we extend the result, first, to deformations of a larger class of symbols with a single singularity on the unit circle, and then to deformations of (almost) general Fisher-Hartwig symbols.