Small maximal clusters are very unlikely in critical random graphs

TL;DR

This paper introduces a probabilistic method using random walk estimates to derive exponential upper bounds on the likelihood of very small maximal components in near-critical random graphs, improving understanding of component size distributions.

Contribution

It provides new exponential bounds for small maximal components in critical random graph models, including near-critical Erdős-Rényi and regular graphs, and extends bounds to intersection and quantum random graphs.

Findings

Exponential upper bounds of order exp(-A^{3/2}) for small components in near-critical Erdős-Rényi graphs.

Improved bounds of order exp(-A^{3/5}) for other critical models like intersection and quantum random graphs.

Optimal bounds for the probability of small maximal components in certain models.

Abstract

We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More specifically, we analyse the near-critical Erd\H{o}s-R\'enyi model $\mathbb{G}(n,p)$ and the random graph $\mathbb{G}(n,d,p)$ obtained by performing near-critical $p$-bond percolation on a simple random $d$-regular graph and show that, for each one of these models, the probability that the size of a largest component is smaller than $n^{2/3}/A$ is at most of order $\exp(-A^{3/2})$. The exponent $3/2$ is known to be optimal for the near-critical $\mathbb{G}(n,p)$ random graph, whereas for the near-critical $\mathbb{G}(n,d,p)$ model the best known upper bound for the above probability was of order $A^{-3/5}$. As a secondary result we show, by means of an optimized version of the martingale method of Nachmias and Peres, that the above probability of observing an unusually small maximal component is at most of order $\exp(-A^{3/5})$ in other two critical models, namely a random intersection graph and the quantum random graph; this stretched-exponential bounds also improve upon the known (polynomial) bounds available for these other two critical models.