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Large deviation principle for the largest eigenvalue of random matrices with a variance profile

Raphaël Ducatez, Alice Guionnet, Jonathan Husson

Abstract

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N \in \mathbb N$, we consider random $N \times N$ symmetric matrices $H^N$ which are such that $H_{ij}^{N}=\frac{1}{\sqrt{N}}X_{i,j}^{N}$ for $1 \leq i,j \leq N$, where the $X_{i,j}^{N}$ for $1 \leq i \leq j \leq N$ are independent and centered. We then denote $Σ_{i,j} ^N = \text{Var} (X_{i,j}^{N}) ( 1 + \textbf{1}_{ i =j})^{-1}$ the variance profile of $H^N$. Our large deviation principle is then stated under the assumption that the $Σ^N$ converge in a certain sense toward a real continuous function $σ$ of $[0,1]^2$ and that the entries of $H^N$ are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving $σ$. This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries.

Large deviation principle for the largest eigenvalue of random matrices with a variance profile

Abstract

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For , we consider random symmetric matrices which are such that for , where the for are independent and centered. We then denote the variance profile of . Our large deviation principle is then stated under the assumption that the converge in a certain sense toward a real continuous function of and that the entries of are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving . This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries.
Paper Structure (25 sections, 50 theorems, 317 equations)

This paper contains 25 sections, 50 theorems, 317 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Limit of the empirical measure of the eigenvalues of $H^{N}$
  4. Main results
  5. Piecewise constant case
  6. Outline of the paper
  7. Notations
  8. Convergence of the annealed spherical integral, Proof of Proposition \ref{['ldpv-1-1']}
  9. Proof of Theorem \ref{['maintheo-constant']}, Large deviation upper bound
  10. Proof of Proposition \ref{['prop:LDPUB2']}
  11. Localization of the spherical integral, Proof of Proposition \ref{['localem-1']}
  12. Tilted probability on the sphere, Proof of Proposition \ref{['prop:Gaussian-Concentration-1']}
  13. Concentration for the spectral measures, proof of Theorem \ref{['prop:G-measure-concentration']}
  14. Adding the condition $\langle\psi,S\psi\rangle>\epsilon$
  15. Proof of Theorem \ref{['maintheo-constant']}, Large deviation lower bound
  16. ...and 10 more sections

Key Result

Lemma 1.1

There exists a family of probability measures $(\mu_{\sigma,t})_{t\in[0,1]}$ supported on a compact interval $[l_{\sigma},r_{\sigma}]\subset\mathbb{R}$ so that if we set we have

Theorems & Definitions (85)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Proposition 1.6
  • Lemma 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Lemma 2.1
  • ...and 75 more