Upper tail of the spectral radius of sparse Erd\H{o}s-R\'{e}nyi graphs

TL;DR

This paper investigates the large deviation probabilities of the largest eigenvalue in sparse Erdős-Rényi graphs, revealing phase transitions and typical structures conditioned on these rare events across different regimes of edge probability.

Contribution

It characterizes the upper tail large deviations of the spectral radius in sparse Erdős-Rényi graphs, identifying phase transitions and the typical structures conditioned on these deviations.

Findings

For p ≫ n^{-2/3}, upper tail probability matches that of planting a clique.

For p ≪ n^{-2/3}, upper tail probability is driven by high degree vertices.

The large deviation probabilities are asymptotically described by a mean-field variational problem.

Abstract

We consider an Erd\H{o}s-R\'{e}nyi graph $\mathbb{G}(n,p)$ on $n$ vertices with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np \le n^{1/2-o(1)}, \label{eq:abs} \tag{$\dagger$} \] and derive the upper tail large deviations of $\lambda(\mathbb{G}(n,p))$, the largest eigenvalue of its adjacency matrix. Within this regime we show that, for $p \gg n^{-2/3}$ the $\log$-probability of the upper tail event of $\lambda(\mathbb{G}(n,p))$ equals to that of planting a clique of an appropriate size (upon ignoring smaller order terms), while for $p \ll n^{-2/3}$ the same is given by that of the existence of a high degree vertex. This, in particular, shows an emergence of {\em non-planted localized structure} in the latter regime. We also confirm that in the entire regime \eqref{eq:abs} the large deviation probability is asymptotically approximated by the solution of the mean-field variational problem, and further identify the typical structure of $\mathbb{G}(n,p)$ conditioned on the upper tail event of $\lambda(\mathbb{G}(n,p))$ in a certain sub-regime of $p$. For $p$ such that $\log(np) \gtrsim \log n$ the large deviations of $\lambda(\mathbb{G}(n,p))$ is deduced from those of ${\rm Hom}(C_{2t}, \mathbb{G}(n,p))$, the homomorphism counts of cycle of length $2t$, for $t \ge 3$ and $p$ such that $n^{1/2-o(1)} \ge np \gg n^{1/t}$. In this latter regime the typical structure of $\mathbb{G}(n,p)$ conditioned on the upper tail of ${\rm Hom}(C_{2t}, \mathbb{G}(n,p))$ is identified and asymptotic tightness of the mean-field approximation is also established.