2–3 Paths in a Lattice Graph: Random Walks

A lattice graph with 2–3 reachability constraints is considered. The graph's vertices are the points with integer nonnegative coordinates in the plane. Each vertex has two outgoing edges, one entering its immediate right neighbor and the other entering its immediate upper neighbor. The admissible paths for 2–3 reachability are those in which the numbers of edges in all but the last inclusion-maximal straight-line segments are divisible by $2$ for horizontal segments and by $3$ for vertical segments. A formula for the number of 2–3 paths from a vertex to a vertex is obtained. A random walk process on the 2–3 paths in the lattice graph is considered. It is proved that this process can locally be reduced to a Markov process on subgraphs determined by the type of the initial vertex. Formulas for the probabilities of vertex-to-vertex transitions along 2–3 paths are obtained.