Asymptotics in bond percolation on expanders

We consider supercritical bond percolation on a family of high-girth d-regular expanders. Alon, Benjamini and Stacey (2004) established that its critical probability for the appearance of a linearsized (“giant”) component is pc = 1/(d − 1). Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant at any p > pc, as well as that of its 2-core. It was further shown in [1] that the second largest component, at any 0 < p < 1, has size at most n for some ω < 1. We show that, unlike the situation in the classical Erdős–Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size n ′ for ω′ arbitrarily close to 1. Moreover, as a by-product of that construction, we answer negatively a question of Benjamini (2013) on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, e.g., the existence of a linear path.