Abstract
We study dynamic random conductance models on Z2 in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.
Highlights
A graph is said to have the infinite collisions property if two independent random walks started at the same location collide infinitely often almost surely
For bounded degree graphs that are not transitive, the infinite collisions property is strictly stronger than recurrence
While it is easy to see that bounded degree transient graphs cannot have infinite collisions, Krishnapur and Peres [30] showed that there exist bounded degree graphs, including the infinite comb graph, that are recurrent but which do not have the infinite collisions property
Summary
A graph is said to have the infinite collisions property if two independent random walks started at the same location collide (occupy the same location at the same time) infinitely often almost surely. Invariance principles are known in the ergodic setting in the non-elliptic case with rates bounded from above (and 0 only on intervals with lengths of finite expectation) [14], and with elliptic rates under moment conditions on the conductances and their reciprocals [6] Such environments need not be reversible, so there exist examples that satisfy (A2) but not (A1). Η has the infinite collisions property almost surely: If X and Y are two random walks on η, both started from the origin at time zero, that are conditionally independent given η, the set {n ∈ N : Xn = Yn} has infinite cardinality and the set {t ∈ [0, ∞) : Xt = Yt} has infinite Lebesgue measure almost surely.
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