The range of a random walk on a comb

Abstract

The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of the neighbors of v, each with probability 1/d(v), where d(v) denotes the degree of v. We answer a question of Cs\'aki, Cs\"org\"o, F\"oldes, R\'ev\'esz, and Tusn\'ady by showing that the expected number of vertices visited by a random walk on the comb after n steps is (1/(2\sqrt{2\pi})+o(1))\sqrt n\log n. This contradicts a claim of Weiss and Havlin.