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

2024/03/08 by Ducatez, Raphaël, Guionnet, Alice, Husson, Jonathan · 2 citations
#60B20 #60F10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2403.05413

Abstract

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For N ∈ \mathbb N, we consider random N × N symmetric matrices H^N which are such that H_ij^N=(1)/(√(N))X_i,j^N for 1 ≤ i,j ≤ N, where the X_i,j^N for 1 ≤ i ≤ j ≤ N are independent and centered. We then denote Σ_i,j ^N = Var (X_i,j^N) ( 1 + 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.

Cited by

• LDP for the largest eigenvalue of Kronecker random matrices
• Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

Related

• Large deviations for the largest eigenvalue of Rademacher matrices
• Large deviations for the largest eigenvalue of generalized sample covariance matrices
• Large Deviations for Random Matrices
• Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails
• LDP for the largest eigenvalue of Kronecker random matrices