Fisher-Hartwig asymptotics for non-Hermitian random matrices
Probability
2026-01-13 v2 Mathematical Physics
Classical Analysis and ODEs
Functional Analysis
math.MP
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.
Cite
@article{arxiv.2512.09123,
title = {Fisher-Hartwig asymptotics for non-Hermitian random matrices},
author = {Paul Bourgade and Guillaume Dubach and Lisa Hartung and Ahmet Keles},
journal= {arXiv preprint arXiv:2512.09123},
year = {2026}
}