Asymptotic Behaviour of the Simple Random Walk on the 2-dimensional Comb

Abstract

We analyze the differences between the horizontal and the vertical component of the simple random walk on the 2-dimensional comb. In particular we evaluate by combinatorial methods the asymptotic behaviour of the expected value of the distance from the origin, the maximal deviation and the maximal span in n steps, proving that for all these quantities the order is n 1/4 for the horizontal projection and n 1/2 for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by n 1/4 and n 1/2 the horizontal and vertical ones, respectively. The limit process is obtained. As a corollary of the estimate of the expected value of the maximal deviation, the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb.

Some Results and Problems for Anisotropic Random Walks on the Plane

“…For random walk on comb we refer to Weiss and Havlin [29], Bertacchi and Zucca [2] and references given there. The following result on weak convergence was established by Bertacchi [1].…”

About the distance between random walkers on some graphs

“…A further insight to the nature of the random walk on a comb was provided by Bertacchi [2], who established the following remarkable weak convergence result for the walk C(n) = (C 1 (n), C 2 (n)) on the comb C 2 .…”

“…Now, just like as it is used in [2], we may apply the Hardy-Littlewood-Karamata theorem in the following form: If…”

The Range of a Random Walk on a Comb

“…This is a rather surprising phenomenon, in view of the fact that the random walk is recurrent, that is, a single random walker visits each site an infinite number of times with probability 1. Some insight was provided by Bertacchi and Zucca [BeZ03] and by Bertacchi [Be06], whose asymptotic estimates suggested that a walker spends most of her time moving vertically along a "tooth" of the comb. Several strong approximation and limit theorems for random walks on a comb have been established by Csáki, Csörgő, Földes, and Révész [CsCs09], [CsCs11].…”

“…According to the main result in [WeH86], the expected value of V n is asymptotically proportional to n 3/4 , for large n. It is not hard to see that almost surely the deviation of the horizontal projection X n of the walk is roughly n 1/4 , while the expected length of the vertical projection is of order n 1/2 . See, e.g., Bertacchi [Be06] (cp. Panny and Prodinger [PaP85]).…”