Fisher-Hartwig asymptotics for non-Hermitian random matrices

TL;DR

This paper extends Fisher-Hartwig asymptotics to two-dimensional non-Hermitian random matrices, revealing their connection to Gaussian multiplicative chaos, logarithmic fields, and Liouville quantum gravity, with implications for universality and phase transitions.

Contribution

It proves a 2D Fisher-Hartwig asymptotic formula for general symbols and applies it to analyze complex spectra of random normal matrices, establishing links to quantum gravity and phase transitions.

Findings

Characteristic polynomial converges to Gaussian multiplicative chaos

Electric potential converges to a logarithmic field

Free energy exhibits a freezing transition

Abstract

We prove the two-dimensional analogue of the asymptotics for Toeplitz determinants with Fisher-Hartwig singularities, for general real symbols. This formula has applications to random normal matrices with complex spectra: (i) the characteristic polynomial converges to a Gaussian multiplicative chaos random measure on the limiting droplet, in the subcritical phase; (ii) the electric potential converges pointwise to a logarithmically correlated field; (iii) the measure of its level sets (i.e. thick points) is identified; (iv) the associated free energy undergoes a freezing transition. This establishes emergence of the Liouville quantum gravity measure from free fermions in 2d, and universality with respect to the external potential.