Dependent percolation and colliding random walks

Abstract

Let G be a connected, undirected graph and X=X0X1X2⋅⋅⋅ and Y=Y0Y1Y2⋅⋅⋅ two simple random walks on G. Let ℕ□ℕ be the nonnegative quadrant of the plane grid, and H the subgraph of ℕ□ℕ induced by the sites (i, j) for which Xi≠Yj. We say that G is “navigable” if with probability greater than 0, the origin belongs to an infinite component of H. We determine which finite graphs are navigable, in particular that K4, the complete graph on four nodes, is navigable but K3 is not. Navigability of G is equivalent to the statement that with positive probability, two tokens taking random walks on G can be moved forward and backward along their paths, and ultimately advanced arbitrarily far, without colliding. The problem is generalized to finite-state Markov chains, and a complete characterization of navigable chains is given. Similar results have been obtained simultaneously and independently by Balister, Bollobás and Stacey, using different methods; our classification theorem relies on a surprising diamond lemma which may be of independent interest. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 58–84, 2000