UC Riverside

Edge percolation on finite transitive graphs is studied analytically and numerically. The results are made rigorous by considering a sequence of finite graphs $(\mathcal{G}_t)_t$ covered by an infinite graph $\mathcal{H}$, and weakly convergent to $\mathcal{H}$. The covering maps are used to classify $1$-cycles on graphs $\mathcal{G}_t$ as homologically trivial or nontrivial, and to define several thresholds associated with the rank of the first homology group on the open subgraphs. The growth of the homological distance $d_t$, the smallest size of a non-trivial cycle on $\mathcal{G}_t$, is identified as the main factor determining the location of homology changing thresholds. In particular, the giant cycle erasure threshold $p_E^0$ (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold $p_c(\mathcal{H})$ if the ratio $d_t/\ln n_t$ diverges, where $n_t$ is the number of edges of $\mathcal{G}_t$, and evidence is given that $p_E^0 < p_c(\mathcal{H})$ in several cases where this ratio remains bounded, which is necessarily the case if $\mathcal{H}$ is non-amenable.

Numerically, finite graphs are constructed with up to $10^5$ edges from several families of locally-planar graphs covered by infinite transitive planar graphs parameterized by Schl\"afli symbols $\{f,d\}$ with $fd/(f+d)\ge 2$, where $d$ regular $f$-gonal faces meet in each vertex. Specifically, considered are the planar regular tiling $\{4,4\}$, regular hyperbolic tilings $\{3,7\}$, $\{3,8\}$, $\{4,5\}$, $\{4,6\}$, $\{5,5\}$, and $\{5,6\}$, their duals with $f$ and $d$ interchanged, as well as random graphs of degree $d=5$---this latter family converges to an infinite tree of the same degree. Conventional and homological percolation are analyzed in these graphs, and the accuracy of several finite-size scaling methods designed calculate the cycle erasure threshold $p_E^0$, the conventional percolation threshold $p_c(\mathcal{H})$, and the giant cluster threshold $p_G$ compared. In particular, the cluster ratio method, one of the most commonly used techniques to locate percolation thresholds, shows rather poor convergence for hyperbolic graphs of this type.