| Abstract: | We consider supercritical bond percolation on a family of high‐girth  ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is  . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any  . It was further shown in the previous study that the second largest component, at any  , has size at most  for some  . We show that, unlike the situation in the classical Erd?s‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size  for  arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini 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, for example, the existence of a linear path. |
