The Range of a Random Walk on a Comb

Pach, János and Tardos, Gábor (2013) The Range of a Random Walk on a Comb. ELECTRONIC JOURNAL OF COMBINATORICS, 20 (3). ISSN 1077-8926

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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 Csaki, Csorgo, Foldes, Revesz, and Tusnady by showing that the expected number of vertices visited by a random walk on the comb after n steps is (1/2 root 2 pi + o(1)) root n log n. This contradicts a claim of Weiss and Havlin.

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