Toeplitz determinants with a one-cut regular potential and Fisher--Hartwig singularities I. Equilibrium measure supported on the unit circle

TL;DR

This paper derives new formulas for Toeplitz determinants with complex symbols involving a regular potential and Fisher--Hartwig singularities, extending known results and applying them to a generalized circular unitary ensemble.

Contribution

It provides novel formulas for Toeplitz determinants with a regular potential supported on the unit circle, including cases with Fisher--Hartwig singularities and non-constant potentials.

Findings

Formulas reduce to known results for constant potentials.

Results are new for non-constant potentials without Fisher--Hartwig singularities.

Applications include statistical properties of a generalized circular unitary ensemble.

Abstract

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher--Hartwig singularities, and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher--Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher--Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.