Fisher-Hartwig asymptotics for non-Hermitian random matrices

Paul Bourgade; Guillaume Dubach; Lisa Hartung; Ahmet Keles

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arXiv · 2512.09123

Fisher-Hartwig asymptotics for non-Hermitian random matrices

Paul Bourgade·Guillaume Dubach·Lisa Hartung·Ahmet Keles

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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.

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BibTeX RIS

Paul Bourgade, Guillaume Dubach, Lisa Hartung, Ahmet Keles. 2025-12-09. Fisher-Hartwig asymptotics for non-Hermitian random matrices. https://arxiv.org/abs/2512.09123

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