Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices

Abstract

Contents 1. Introduction 2. Ideas of the proof 3. Large deviation upper bound 4. Quenched asymptotics for spherical integrals 5. Large deviation lower bound 6. The top-eigenvalue of tilted Wigner matrices 7. Constrained Gibbs variational principle 8. Annealed asymptotics for restricted spherical integrals 9. Proof of Theorem 1.8 10. Proof of Corollary 1.12 11. Proof of Theorem 1.13 12. Proof of Theorem 1.14 Appendix A. Concentration properties for sub-Gaussian Wigner matrices Appendix B. Ruling out localized eigenvectors Appendix C. Coupling of tilted laws Appendix D. Proof of Lemma 8.4 ReferencesAbstract. We establish precise estimates for the probability of rare events of the largest eigenvalue of Wigner matrices with sub-Gaussian entries. In contrast to the case of Wigner matrices with heavier tails, where deviations are governed by the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that in the general sub-Gaussian case that deviations can be caused by a mixture of localized and delocalized changes in the entries. Our key result is a finite-N approximation for the probability of rare events by an optimization problem involving restricted annealed free energies for a spherical spin glass model. This allows us to derive full large deviation principles for the largest eigenvalue in several cases, including when the law of the matrix entries is compactly supported and symmetric, as well as the case of randomly sparsified GOE matrices.

Spectral large deviations of sparse random matrices

“…Together with Augeri [9], they also showed that for a more general class of sub-Gaussian matrices, the rate function is universal for small large deviations, but beyond that also depends on other properties of the moment generating function. Subsequently, continuing this line of work, Cook, Ducatez and Guionnet, in [21], strengthened the result into a full large deviation principle for such sub-Gaussian Wigner matrices.…”

“…An important distinction from denser matrices is that the universality of the spectral behavior breaks down in the sparse case and the spectrum depends rather crucially on the entry distribution. While the study of the bulk spectral properties of sparse random matrices has witnessed some activity, for example, [15,16], the precise edge statistics has still been mostly out of reach of the known methods which are primarily tailored to analyze denser graphs, see for instance [9,21].…”