Phase transition of disordered random networks on quasi-transitive graphs

Abstract

Given a quasi-transitive infinite graph $G$ with volume growth rate ${\rm gr}(G),$ a transient biased electric network $(G,\, c_1)$ with bias $\lambda_1\in (0,\,{\rm gr}(G))$ and a recurrent biased one $(G,\, c_2)$ with bias $\lambda_2\in ({\rm gr}(G),\infty).$ Write $G(p)$ for the Bernoulli-$p$ bond percolation on $G$ defined by the grand 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 $c_2(g)$. Our main results are as follows: (i) On connected quasi-transitive infinite graph $G$ with percolation threshold $p_c\in (0,\, 1),$ $(G,\, c_1,\, c_2,\, p)$ has a non-trivial recurrence/transience phase transition such that the threshold $p_{c}^{*}\in (0,\, 1)$ is deterministic, and almost surely $(G,\, c_1,\, c_2,\, p)$ is recurrent for $p<p_c^*$ and transient for $p>p_c^*.$ There is a non-trivial recurrence/transience phase transition for $(G,\, c_1,\, c_2,\, p)$ with $G$ being a Cayley graph if and only if the corresponding group is not virtually $\mathbb{Z}$. (ii) On $\mathbb{Z}^d$ for any $d\geq 1,$ $p_c^{*}= p_c$. And on $d$-regular trees $\mathbb{T}^d$ with $d\geq 3$, $p_c^{*}=(\lambda_1\vee 1) p_c$, and thus $p_c^{*}>p_c$ for any $\lambda_1\in (1,\,{\rm gr}(\mathbb{T}^d)).$ As a contrast, we also consider phase transition of having unique currents or not for $(\mathbb{Z}^d,\, c_1,\, c_2,\, p)$ with $d\geq 2$ and prove that almost surely $(\mathbb{Z}^2,\, c_1,\, c_2,\, p)$ with $\lambda_1<1\leq\lambda_2$ has unique currents for any $p\in [0,1]$.