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Fisher-Hartwig asymptotics for non-Hermitian random matrices

Paul Bourgade, Guillaume Dubach, Lisa Hartung, Ahmet Keles

TL;DR

This work develops a full 2d Fisher–Hartwig theory for determinants of non-Hermitian random matrices with singular potentials, establishing precise asymptotics that connect to Gaussian multiplicative chaos on the limiting droplet. The authors combine Ward identities with a surgical decoupling of local singularities and a Ginibre/Kostlan comparison to prove emergence of GMC measures and log-correlated electric fields, revealing universality with respect to the confining potential. They provide explicit moment formulas, CLTs for linear statistics, and a freezing transition in the subcritical GMC regime, demonstrating a deep link between 2d free fermions, random normal matrices, and Liouville quantum gravity. The results unify 2d FH-type asymptotics with GMC limits, improving understanding of thick points, extreme values, and phase transitions in non-Hermitian random matrix models. The methods pave the way for further exploration of Painlevé-type phenomena and universality across general confining potentials.

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.

Fisher-Hartwig asymptotics for non-Hermitian random matrices

TL;DR

This work develops a full 2d Fisher–Hartwig theory for determinants of non-Hermitian random matrices with singular potentials, establishing precise asymptotics that connect to Gaussian multiplicative chaos on the limiting droplet. The authors combine Ward identities with a surgical decoupling of local singularities and a Ginibre/Kostlan comparison to prove emergence of GMC measures and log-correlated electric fields, revealing universality with respect to the confining potential. They provide explicit moment formulas, CLTs for linear statistics, and a freezing transition in the subcritical GMC regime, demonstrating a deep link between 2d free fermions, random normal matrices, and Liouville quantum gravity. The results unify 2d FH-type asymptotics with GMC limits, improving understanding of thick points, extreme values, and phase transitions in non-Hermitian random matrix models. The methods pave the way for further exploration of Painlevé-type phenomena and universality across general confining potentials.

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.

Paper Structure

This paper contains 32 sections, 37 theorems, 274 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fisher-Hartwig asymptotics.
  3. Gaussian multiplicative chaos.
  4. Outline.
  5. Assumptions on the potential.
  6. Notations and conventions.
  7. Scales and geometry.
  8. Function classes and differential operators.
  9. Cutoff functions and log-regularization.
  10. Biased measures.
  11. Barnes $\rm{G}$-function.
  12. Preliminaries on nonsingular potentials
  13. Ward identity.
  14. Harmonic extension and decomposition.
  15. Application of the Ward identity.
  16. ...and 17 more sections

Key Result

Theorem 1

For any fixed $m$, distinct $\zeta_j$ in the interior of $\mathbb{D}$ and positive exponents $\gamma_j$, $1\leqslant j\leqslant m$, we have where ${\rm G}$ is the Barnes ${\rm G}$-function. Such non-Hermitian Fisher-Hartwig asymptotics -- in the full form stated in Subsection subsec:FH -- imply the following convergence on $\mathbb{D}$, in distribution with respect to the weak topology and in the

Figures (1)

  • Figure 1: The random field $\frac{|\det(G-z)|}{\mathbb{E}{|\det(G-z)|}}$ for $|z|<1$ and a Ginibre matrix $G$ of size $N=1000$.

Theorems & Definitions (68)

  • Theorem
  • Definition 1.1
  • Theorem 1.2: General potential
  • Theorem 1.3: Ginibre ensemble
  • Remark 1.4: Complex exponents
  • Remark 1.5: Jump singularities
  • Remark 1.6: Painlevé transcendents
  • Corollary 1.7: Log-correlations pointwise
  • Theorem 1.8: Convergence to a GMC
  • Remark 1.9: Outside the droplet
  • ...and 58 more