A discrete log gas, discrete Toeplitz determinants with Fisher-Hartwig singularities, and Gaussian Multiplicative Chaos

TL;DR

This paper analyzes a discrete log-gas on the unit circle, showing that under certain conditions, the characteristic polynomial's absolute value relates to Gaussian multiplicative chaos, and confirming the Fisher-Hartwig conjecture for discrete Toeplitz determinants.

Contribution

It establishes a connection between discrete log-gases and Gaussian multiplicative chaos, and proves the Fisher-Hartwig conjecture for discrete Toeplitz determinants under specific density conditions.

Findings

Characteristic polynomial relates to Gaussian multiplicative chaos in sparse regimes.

Fisher-Hartwig conjecture holds for discrete Toeplitz determinants in certain density regimes.

Denser gases may require modified formulas for the characteristic polynomial.

Abstract

We consider a log-gas on a discretization of the unit circle. We prove that if the gas is not too dense, or the number of particles in the gas is not too large compared to the scale of the discretization, the absolute value of the characteristic polynomial can be described in terms of a Gaussian multiplicative chaos measure. This is done by analyzing discrete Toeplitz determinants with Fisher-Hartwig singularities. In particular, we prove that if the gas is not too dense, the classical Fisher-Hartwig conjecture holds for the discrete Toeplitz determinant as well. Our analysis suggests that if the gas is any denser than this, the formula needs to be modified.