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Quenched universality for deformed Wigner matrices

Giorgio Cipolloni, László Erdős, Dominik Schröder

Abstract

Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix $H$ yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices $H+xA$ with a deterministic Hermitian matrix $A$ and a fixed Wigner matrix $H$, just using the randomness of a single scalar real random variable $x$. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.

Quenched universality for deformed Wigner matrices

Abstract

Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices with a deterministic Hermitian matrix and a fixed Wigner matrix , just using the randomness of a single scalar real random variable . Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.

Paper Structure

This paper contains 8 sections, 4 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Monoparametric universality via eigenbasis rotation
  4. Monoparametric universality via spectral sampling
  5. Gaudin-Mehta distribution
  6. Quenched universality: Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm2b']}
  7. Universality via eigenbasis rotation mechanism: Proof of Theorem \ref{['thm1']}
  8. Universality via spectral sampling mechanism: Proof of Theorem \ref{['thm2b']}

Key Result

Corollary \oldthetheorem

Let $H$ be a Wigner matrix and $I\subset (-2+\epsilon,2-\epsilon)$ be an interval in the bulk of $H$ of length $\lvert I\rvert\ge N^{-1+\xi}$ for some $\epsilon,\xi>0$. Then there exist small $\kappa, \alpha>0$ and an event $\Omega_I$ in the probability space $\mathop{\mathrm{\mathbf{P}}}\nolimits_H where the implicit constant in univ2 and $\kappa,\alpha$ depend on $\epsilon, \xi$.

Figures (2)

  • Figure 1: The two types of universality: The first histogram shows the normalised gaps of the two middle eigenvalues in the spectrum of $5000$ complex Wigner matrices of size $100\times 100$. The second histogram shows the empirical normalised bulk eigenvalue gaps of a single complex Wigner matrix of size $5000\times 5000$. Both distributions asymptotically approach the Gaudin-Mehta distribution$p_2$ drawn as solid lines, see Section \ref{['sec painleve']}.
  • Figure 2: Here we show the histogram of a (rescaled) single eigenvalue gap $\lambda_{N/2+1}-\lambda_{N/2}$ in the middle of the spectrum for $N\in\set{2,100,1000}$ and for random matrices sampled from either the GUE or the monoparametric ensemble. For the GUE ensemble the histogram has been generated by sampling $2000$ independent GUE matrices $H$. For the monoparametric ensemble only two GUE matrices $H,A$ have been drawn at random, and the histogram has been generated by sampling $2000$ standard Gaussian random variables $x$ and considering the gaps of $H^x=H+xA$. The solid black line represents the theoretical limit $p_2(s)$ which matches the empirical distribution very closely already for $N=2$. In Appendix \ref{['appendix figure']} we present numerical evidence for the speed of convergence, inspired by the observation on the slow convergence of the spectral form factor made in GPSW.

Theorems & Definitions (8)

  • Corollary \oldthetheorem: to Theorem \ref{['thm2b']}
  • Theorem \oldthetheorem: Quenched universality for monoparametric ensemble
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem: Quenched monoparametric universality via spectral sampling mechanism
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • proof : Proof of Theorem \ref{['thm1']}