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The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated

Krishnan Mody

Abstract

We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum.

The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated

Abstract

We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum.
Paper Structure (14 sections, 41 theorems, 239 equations)

This paper contains 14 sections, 41 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Model Assumptions.
  3. Main Results.
  4. Outline of the paper.
  5. Preliminaries
  6. Wegner Estimate at the Edge
  7. Local Law
  8. Smoothing
  9. Coupling of Determinants
  10. Proof of Theorem \ref{['thm:log_corr_field']}
  11. Appendix A: Mesoscopic Central Limit Theorem
  12. Model Assumptions.
  13. Preliminary Estimates.
  14. Calculation of the Characteristic Function.

Key Result

Theorem 1.2

For fixed ${\varepsilon}, c > 0$, define the subset of the spectrum, and for fixed $m \geqslant 1$, let $(E_1,\dots,E_m)_{N \geqslant 1} \in \mathcal{G}_{{\varepsilon}, c}^m$, possibly depending on $N$. Let and assume that the limits, exist. Denote ${\bf{a}}=(a_{ij})_{1 \leqslant i,j \leqslant m}$ and ${\bf{b}} =(b_{ij})_{1 \leqslant i,j \leqslant m}$, and write $L_N(E) = L_N(E, {\bm{\lambda}})

Theorems & Definitions (73)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 2.1: Local Law
  • Corollary 2.2: Rigidity
  • ...and 63 more