Spectral large deviations of sparse random matrices

Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean $p$. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph $G_{n,p}$ equipped with independent and identically distributed (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. While certain techniques have been devised for the case when $p$ is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case $p=\frac{d}{n}$, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph $G_{n,\frac{d}{n}}$ without edge-weights and with Gaussian edge-weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions 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 the law of large numbers behavior for the largest eigenvalue, which also seems to be new and difficult to obtain using existing methods. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at $\alpha =2$, that is, corresponding to the Gaussian distribution.