Upper tail of the spectral radius of sparse Erdős-Rényi graphs

Abstract

We consider an Erdős-Rényi graph G(n, p) on n vertices with edge probability p such that log n log log n ≪ np n 1/2−o(1) , and derive the upper tail large deviations of the largest eigenvalue of the adjacency matrix. Within this regime we show that, for p ≫ n −2/3 the log-probability of the upper tail of the largest eigenvalue equals to that of planting a clique of an appropriate size (upon ignoring smaller order terms), while for p ≪ n −2/3 the same is given by that of the existence of a high degree vertex.In the regime log(np) log n the large deviations of the spectral radius is derived from that of homomorphism counts of cycles of even length. A similar dichotomy, as above, arises for the homomorphism counts of cycles of length 2t, with the threshold depending on t.This, in particular, identifies a sparsity regime, both in the context of the largest eigenvalue and homomorphism counts of even cycles, where the large deviation is due to the presence of a non-planted localized structure. It further demonstrates that in a certain sparsity regime the large deviation rate functions for the homomorphism and subgraph counts differ from each other, while the large deviation speeds continue to be the same, even when the expected value of the former is 1 + o(1) times that of the latter.Our results also identify the typical behavior of G(n, p) conditioned on the upper tail events of the spectral radius and homomorphism counts of even cycles in certain sub-regimes of p.

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

“…In general, we say that large deviations for a function of d independent random variables exhibit a localization phenomenon when the driving mechanism for the deviation involves a deviation of o(d) variables from their typical ranges. Apart from the results of [3,18] for Wigner matrices with stretched exponential tails, localization phenomena have been shown in recent years to govern large deviations for the extreme eigenvalues of adjacency matrices for sparse random graphs [7,8,15,16,[20][21][22]37] and random networks [32,33].…”