Random Walks on graphs, electric networks and fractals

Abstract

This paper is devoted to the problem of the nearest, neighbour Random Walk on infinite graphs. We investigate the RW Xn started from a fixed vertex X0=x∈V of the graph G=(V, E) and the expected value of the first exit time TN from the N-ball BN in G. It will be shown that if G is sufficiently "regular" then {Mathematical expression} {Mathematical expression} and {Mathematical expression} where dR is the RW dimension, d is the fractal dimension and dΩ is the exponent of the growth of the resistance of BN. Though the method of the paper was developed for the case when X0=x is a fixed vertex of the graph, we hope that the result can be generalized. © 1989 Springer-Verlag.