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Critical edge statistics for deformed GinUEs

Dang-Zheng Liu, Lu Zhang

TL;DR

This work identifies the complete edge-statistics dichotomy for the deformed complex Ginibre ensemble: standard GinUE edge statistics at regular edge points and a novel, Pearcey-type critical edge statistics at critical boundary points. The authors develop a intricate integral-representation framework, reduce the problem via translation, translation-invariant symmetries, and concentration arguments, and implement a seven-step Taylor expansion alongside determinant expansions to isolate the leading behavior. The main achievement is a precise description of the critical-edge $n$-point correlations in terms of a matrix integral $I_n(\hat Z)$, combined with explicit exponential and determinant prefactors, providing a non-Hermitian analogue of Pearcey universality and suggesting a third universal edge class in non-Hermitian random matrix theory. The results illuminate the delicate interplay between spectral geometry of the perturbation $X_0$, the Brown measure boundary behavior, and non-Hermitian local statistics, and pave the way for further exploration of universal edge phenomena beyond the Ginibre family.

Abstract

For the complex Ginibre ensemble subjected to an additive perturbation by a deterministic normal matrix $X_0$, we establish that under specific spectral conditions on $X_0$, only two distinct types of local spectral statistics emerge at the spectral edge: GinUE statistics and critical statistics, which respectively correspond to regular and quadratically vanishing spectral points. The critical statistics, as a non-Hermitian analogue of Pearcey statistics in random matrix theory, describes a novel point process on the complex plane. This identifies the third (and likely final) universal statistics in non-Hermitian random matrix theory, after the established GinUE bulk and edge universality classes, and represents the primary achievement of this paper.

Critical edge statistics for deformed GinUEs

TL;DR

This work identifies the complete edge-statistics dichotomy for the deformed complex Ginibre ensemble: standard GinUE edge statistics at regular edge points and a novel, Pearcey-type critical edge statistics at critical boundary points. The authors develop a intricate integral-representation framework, reduce the problem via translation, translation-invariant symmetries, and concentration arguments, and implement a seven-step Taylor expansion alongside determinant expansions to isolate the leading behavior. The main achievement is a precise description of the critical-edge -point correlations in terms of a matrix integral , combined with explicit exponential and determinant prefactors, providing a non-Hermitian analogue of Pearcey universality and suggesting a third universal edge class in non-Hermitian random matrix theory. The results illuminate the delicate interplay between spectral geometry of the perturbation , the Brown measure boundary behavior, and non-Hermitian local statistics, and pave the way for further exploration of universal edge phenomena beyond the Ginibre family.

Abstract

For the complex Ginibre ensemble subjected to an additive perturbation by a deterministic normal matrix , we establish that under specific spectral conditions on , only two distinct types of local spectral statistics emerge at the spectral edge: GinUE statistics and critical statistics, which respectively correspond to regular and quadratically vanishing spectral points. The critical statistics, as a non-Hermitian analogue of Pearcey statistics in random matrix theory, describes a novel point process on the complex plane. This identifies the third (and likely final) universal statistics in non-Hermitian random matrix theory, after the established GinUE bulk and edge universality classes, and represents the primary achievement of this paper.
Paper Structure (16 sections, 8 theorems, 324 equations, 1 figure)

This paper contains 16 sections, 8 theorems, 324 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Proof sketch and article structure
  5. Concentration reduction
  6. Notation
  7. Translation reduction
  8. Matrix variables
  9. Integral representation
  10. Localization reduction
  11. Maximum lemmas
  12. Proof of Theorem \ref{['2-complex-correlation critical2']}
  13. Taylor expansion of $f(\boldsymbol{T},Y,\boldsymbol{Q})$
  14. Taylor expansion of $\det(\widehat{L}_1+\sqrt{\gamma_N}\widehat{L}_2)$.
  15. Matrix integrals and final proof
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $R_{N}^{(n)}\left( X_0; z_1,\cdots,z_{n} \right)$ be the $n$-point correlation function for the deformed ensemble ${\mathrm{GinUE}}_{N}(X_0)$ with $\tau=1+N^{-\frac{1}{2}}\sqrt{P_1}\hat{\tau}$. Under the assumption A0 form on $X_0$, let $\chi$ be given in parameter2, if $z_0$ is a critical bound

Figures (1)

  • Figure 1: Plot (a) shows the simulation surface graph of the microscopic rescaled level density $N^{-\frac{1}{4}}R_N^{(1)}( X_0;z_0+N^{-\frac{1}{4}}\hat{z} )$ near the point $z_0=0$ with 5000 samples and $N=3000$, and with $X_{0}=\mathrm{diag}(\mathbb{I}_{N/2}, -\mathbb{I}_{N/2})$. Plot (b) shows the surface graph of the limiting one-point function with $r_0=0$ and $\hat{z}=x+{\rm i}y$, as a limit of Plot (a).

Theorems & Definitions (18)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 8 more