Homology-changing percolation transitions on finite graphs

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 classify 1-cycles Gt as homologically trivial or non-trivial define several thresholds associated with rank thus defined first homology group open subgraphs generated Bernoulli (edge) process. identify growth distance dt, smallest size cycle Gt, main factor determining location homology-changing thresholds. show that giant erasure threshold pE0 (related conventional for corresponding generalized toric codes) coincides pc(H) if ratio dt/ln nt diverges, where is number edges give evidence pE0<pc(H) in cases this remains bounded, which necessarily case non-amenable.