Site percolation on pseudo‐random graphs

We consider vertex percolation on pseudo-random d $$ -regular graphs. The previous study by the second author established existence of phase transition from small components to a linear (in n \frac{n}{d} ) sized component, at p = 1 p=\frac{1}{d} . In supercritical regime, our main result recovers sharp asymptotic size largest and shows that all other are typically much smaller. Furthermore, we typical properties component such as number edges, long cycle expansion. subcritical strengthen upper bound likely size.