Spectral large deviations of sparse random matrices

TL;DR

This paper investigates the large deviations of the largest eigenvalue in sparse random matrices, revealing universal behavior for heavy tails and non-universal, variational characterizations for lighter tails, with implications for spectral graph theory.

Contribution

It extends large deviation analysis to weighted Erdős-Rényi graphs with general tail distributions, uncovering universal and non-universal regimes and connecting to spectral graph theory.

Findings

Universal large deviation behavior for $oldsymbol{ ext{tail decay } e^{-t^eta} ext{ with } eta > 2$.

Non-universal large deviation rate function characterized by a variational problem for $eta < 2$.

Phase transition in the typical largest eigenvalue at $eta=2$, the Gaussian tail case.

Abstract

Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices is formed by the adjacency matrix of an Erd\H{o}s-R\'{e}nyi graph $\mathcal{G}_{n,p}$ equipped with i.i.d. edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. We focus on the case $p = \frac{d}{n}$, where most known techniques break down. So far, results were known only for $\mathcal{G}_{n,\frac{d}{n}}$ without edge-weights (Krivelevich and Sudakov, '03), (Bhattacharya, Bhattacharya, and Ganguly, '21) and with Gaussian edge-weights (Ganguly and Nam, '21). In the present article, we consider the effect of general weight distributions. More specifically, we consider the entries whose tail probabilities decay at rate $e^{-t^\alpha}$ with $\alpha>0$, where the regimes $0<\alpha<2$ and $\alpha>2$ correspond to tails heavier and lighter than the Gaussian tail respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when $\alpha > 2.$ In contrast, in the $\alpha< 2$ case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin-Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish new law of large numbers results for the largest eigenvalue. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at $\alpha = 2$, i.e. the Gaussian distribution.