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Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices

Fanny Augeri

Abstract

Let $Ξ$ be the adjacency matrix of an Erdős-Rényi graph on $n$ vertices and with parameter $p$ and consider $A$ a $n\times n$ centered random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree $np$ diverges, the empirical spectral measure of the normalized Hadamard product $(A \circ Ξ)/\sqrt{np}$ converges weakly in probability to the semicircle law. In the regime where $p\ll 1$ and $ np \gg \log n$, we prove a large deviations principle for the empirical spectral measure with speed $n^2p$ and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale $n^2p$ are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs.

Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices

Abstract

Let be the adjacency matrix of an Erdős-Rényi graph on vertices and with parameter and consider a centered random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree diverges, the empirical spectral measure of the normalized Hadamard product converges weakly in probability to the semicircle law. In the regime where and , we prove a large deviations principle for the empirical spectral measure with speed and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs.
Paper Structure (14 sections, 33 theorems, 193 equations)

This paper contains 14 sections, 33 theorems, 193 equations.

Table of Contents

  1. Introduction and main result
  2. Setup and main result
  3. Related work
  4. Outline of the proof
  5. Notation
  6. An exponential equivalent
  7. A concentration argument
  8. Stability of QVEs
  9. Complexity of the kernels $W_n$ for the cut norm
  10. Large deviation principle for weighted sparse graphs
  11. A counting lemma for trees
  12. Proofs of Theorem \ref{['main']} and Lemma \ref{['zeroI']}
  13. Proof of Theorem \ref{['main']}
  14. Proof of Lemma \ref{['zeroI']}

Key Result

Theorem 1.4

Let $\widehat{X}$ be a sparse Wigner matrix with bounded entries and set $X:=\widehat{X}/\sqrt{np}$. Assume $p\ll 1$ and $np \gg \log n$. The sequence $\mu_X$ satisfies a large deviation principle in $\mathcal{P}(\mathbb{R})$ endowed with the weak topology, with speed $n^2p$ and good rate function $

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: QVE measure of a kernel
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Remark 1.7
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • ...and 56 more