Homology-changing percolation transitions on finite graphs

Abstract

We consider homological edge percolation on a sequence (Gt)t of finite graphs covered by an infinite (quasi)transitive graph H and weakly convergent to H. In particular, we use the covering maps to classify 1-cycles on graphs Gt as homologically trivial or non-trivial and define several thresholds associated with the rank of thus defined first homology group on the open subgraphs generated by the Bernoulli (edge) percolation process. We identify the growth of the homological distance dt, the smallest size of a non-trivial cycle on Gt, as the main factor determining the location of homology-changing thresholds. In particular, we show that the giant cycle erasure threshold pE0 (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold pc(H) if the ratio dt/ln nt diverges, where nt is the number of edges of Gt, and we give evidence that pE0<pc(H) in several cases where this ratio remains bounded, which is necessarily the case if H is non-amenable.