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arXiv · 2002.01010

Large deviations for the largest eigenvalue of matrices with variance profiles

Jonathan Husson
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Abstract

In this article we consider Wigner matrices $X_N$ with variance profiles (also called Wigner-type matrices) which are of the form $X_N(i,j) = σ(i/N,j/N) a_{i,j} / \sqrt{N}$ where $σ$ is a symmetric real positive function of $[0,1]^2$ and $σ$ will be taken either continuous or piecewise constant. We prove a large deviation principle for the largest eigenvalue of those matrices under the same condition of sharp sub-Gaussian bound and for some other assumptions on $σ$. These sub-Gaussian bounds are verified for example for Gaussian variables, Rademacher variables or uniform variables on $[- \sqrt{3}, \sqrt{3}]$.

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Jonathan Husson. 2023-02-11. Large deviations for the largest eigenvalue of matrices with variance profiles. https://doi.org/10.1214/22-ejp793

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