Site percolation on non-regular pseudo-random graphs

Abstract

We study site percolation on a sequence of graphs $\{G_n\}_{n\geq1}$ on $n$ vertices where degree of each vertex is in the interval $(np -a_n, np+a_n)$ and the co-degree of every pair of vertices is at most ${n}p^2+ b_n$, where $p \in (0,1)$ and $\{a_n\}_{n\geq1}$, $\{b_n\}_{n\geq1}$ are sequences of real numbers. Under suitable conditions on $p \in (0,1)$, $a_n$'s and $b_n$'s we show that site percolation on these sequences of graphs undergo a sharp phase transition at $\frac{1}{np}$. More precisely for $\varepsilon>0$, we form a random set $R(\rho_n)$ by including each vertex of $G_n$ independently with probability $\rho_n$. If $\rho_n = \frac{1-\varepsilon}{np}$, then for every small enough $\varepsilon>0$ and $n$ large enough, all connected components in the subgraph of $G_n$ induced by $R(\rho_n)$ are of size at most poly-logarithmic in $n$ with high probability. If $\rho_n = \frac{1+\varepsilon}{np}$, then for every small enough $\varepsilon>0$ and $n$ large enough, the subgraph of $G_n$ induced by $R(\rho_n)$ contains a 'giant' connected component of size at least $\frac{\varepsilon}{p}$ with high probability. Further, we show that under an additional assumption on $\{b_n\}_{n\geq 1}$ the giant component is unique. This partially resolves a question by Krivelevich \cite{krivelevich2016phase} regrading uniqueness of the giant component of site percolation in a general class of regular pseudo-random graphs. We hope that our method of proving uniqueness of the giant component will be applicable in other contexts as well.