Dynamical percolation. (English) Zbl 0894.60098

Häggström, Olle; Peres, Yuval; Steif, Jeffrey E.

Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 4, 497-528 (1997).

Summary: We study bond percolation evolving in time in such a way that the edges turn on and off independently according to a continuous time stationary 2-state Markov chain. Asking whether an infinite open cluster exists for a.e. \(t\) reduces (by Fubini’s theorem) to ordinary bond percolation. We ask whether “a.e. \(t\)” can be replaced by “every \(t\)” and show that for sub- and supercritical percolation the answer is yes (for any graph), while at criticality the answer is no for certain graphs. For instance, there exist graphs which do not percolate at criticality for a.e. \(t\), but do percolate for some exceptional \(t\). We show that for \(\mathbb{Z}^d\), \(d\geq 19\), there is a.s. no infinite open cluster for all \(t\) at criticality. We give a sharp criterion for a general tree to have an infinite open cluster for some \(t\), in terms of the effective conductance of the tree (analogous to a criterion of R. Lyons for ordinary percolation on trees). Finally, we compute the Hausdorff dimension of the set of times for which an infinite open cluster exists on a spherically symmetric tree.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C43	Time-dependent percolation in statistical mechanics

Keywords:

bond percolation; sub- and supercritical percolation; Hausdorff dimension; infinite open cluster