The probability of unusually large components for critical percolation on random d-regular graphs

Abstract

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\ge 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n,d,p)$ be a (random) graph on n vertices obtained by drawing uniformly at random a d-regular (simple) graph on $[n]$ and then performing independent p-bond percolation on it, i.e. we independently retain each edge with probability p and delete it with probability $1-p$. Let $|{\mathcal{C}}_{\text{max}}|$ be the size of the largest component in $\mathbb{G}(n,d,p)$. We show that, when p is of the form $p={(d-1)}^{-1}(1+\mathit{\lambda }{n}^{-1∕3})$ for $\mathit{\lambda }\in \mathbb{R}$, and A is large,

$$\mathbb{P}(|{\mathcal{C}}_{\text{max}}|>A{n}^{2∕3})\asymp A^{-3∕2}{e}^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{\mathit{\lambda }{A}^2(d-1)}{2d}-\frac{{\mathit{\lambda }}^{2}A(d-1)}{2(d-2)}}.$$

This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than $A{n}^{2∕3}$.