Phase transition of disordered random networks on quasi-transitive graphs

Take a quasi-transitive infinite graph G, a transient biased electric network (G, c(1)) with positive bias lambda(1) and a recurrent biased one (G, c(2)) with bias lambda(2) is an element of (lambda 1, infinity). Write G(p) for the Bernoulli-p bond percolation on G, and define the percolation process (G(p))(p is an element of [0, 1]) by the standard coupling. Let (G, c(1), c(2), p) be the following biased disordered random network: Open edges e in G(p) take the conductance c(1)(e), and closed edges g in G(p) take the conductance c(2)(g). Then the following hold: (i) On graph G with percolation threshold p(c) is an element of (0, 1), (G, c(1), c(2), p) has a non-trivial recurrence/transience phase transition such that the threshold p(c)& lowast; is an element of (0, 1) is deterministic, and almost surely, (G, c(1), c(2), p) is recurrent for any p < p(c)* and transient for any p > p(c)*. (ii) For Z(d) (d >= 2) or any Cayley graph G of any group which is virtually Z, p(c)* = p(c); and for d-regular trees T-d with d > 3, p(c)* = pc if lambda(1) <= 1 and p(c)* = lambda(1)pc > pc if 1 < lambda(1) <lambda(c) where lambda(c) is the threshold for recurrence and transience of the biased network(T-d, c(lambda)). (iii) There is no phase transition of having unique currents or not for (Z(2), c(1), c(2), p) with lambda(1) < 1 <= lambda(2), in the sense that almost surely, (Z(2), c(1), c(2), p) has unique currents for any 0 <= p <= 1. Moreover, for a systematic study of ((G, c(1), c(2), p))p is an element of [0,1], several interesting problems and conjectures are proposed.