Generalized percolation games on the two-dimensional square lattice and ergodicity of associated probabilistic cellular automata

Abstract

We consider a highly generalized set-up in which each vertex of the infinite two-dimensional square lattice graph (whose set of vertices is , with each vertex (x, y) adjacent to each of and ) is assigned, independent of all else, a label that reads trap with probability p, target with probability q, and open with the remaining probability , and, in addition, each edge is assigned, independent of all else, a label that reads trap with probability r and open with probability . This model encompasses the seemingly more general model where, in addition to all the vertex-labels and edge-labels described above, an edge can also be labeled as a target, since assigning the label of target to an edge going from (x, y) to either or is equivalent to marking the vertex (x, y) as a trap. A percolation game is played on this random board, involving two players and a token. The players take turns to make moves, where a move involves relocating the token from where it is currently located, say some vertex , to any one of and . A player wins if she is able to move the token to a vertex labeled as a target, or force her opponent to either move the token to a vertex labeled as a trap or along an edge labeled as a trap. We seek to find a regime, in terms of values of the parameters p, q, and r, in which the probability of this game resulting in a draw equals 0. We further consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability r, target with probability s, and open with probability , but the vertices are left unlabeled, and various regimes of values of r and s are explored in which the probability of draw is guaranteed to be 0. We show that the probability of draw in each such game equals 0 if and only if a suitably defined probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions or potential functions to investigate the regimes in which said PCA is ergodic. We mention here that one of the main results of Holroyd et al. (2019 Probab. Theory Related Fields 174, 1187–1217) follows as a special case of our main result. Moreover, our result shows that a phase transition happens at the origin (i.e. at in the case of generalized percolation games, and at in the case of bond percolation games) in the sense that, the probability of draw equals 1 at (respectively, at ), whereas in every neighborhood around (0, 0, 0) (respectively, (0, 0)), there exists some value of (p, q, r) (respectively, (r, s)) for which the probability of draw equals 0.