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 n1/4 for the horizontal projection and n1/2 for the vertical one (the exact constants are determined). Then we rescale the two projections of the random walk dividing by n1/4 and n1/2 the horizontal and vertical ones, respectively. The limit process is obtained. With similar techniques the walk dimension is determined, showing that the Einstein relation between the fractal, spectral and walk dimensions does not hold on the comb. © 2006 Applied Probability Trust.