Site percolation on non-regular pseudo-random graphs

We study site percolation on a sequence of graphs $\{G_n{\}}_{n\ge 1}$ 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 $\epsilon >0$, we form a random set $R({\rho }_n)$ by including each vertex of $G_n$ independently with probability ${\rho }_n$. If ${\rho }_n=\frac{1-\epsilon }{np}$, then for every small enough $\epsilon >0$ 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 $\rho_n = \frac{1+\varepsilon}{np}$,thenforeverysmallenough$\varepsilon>0$and$n$largeenough,thesubgraphof$G_n$inducedby$R(\rho_n)$ contains a ‘giant’ connected component of size at least $\frac{\epsilon }{p}$ with high probability. Further, we show that under an additional assumption on $\{b_n{\}}_{n\ge 1}$ 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.