Symmetric Random Walks on Groups

Let G he a countable group and let A = {au ax, } (diCG) generate G. Consider the random walk on G in which every step consists of right multiplication by a,-or its inverse ar1, each with probability pi (pi = 0, 2 JZi pi= 1). This does not mean that p,-is the total probability of multiplying by any element which equals a,-in G. It may be, for instance, that ai = aj with j^i (or ai = aT1). In this case the total probability of multiplying by at is at least pi+pj (resp. 2pj). We say that P = {pi, p2, } is a probability distribution on the set of generators A. This random walk defines a Markov chain whose possible states are the elements of G. The transition probability from gi to g2 (g, g2CG) is given by the probability that g2 is reached in one step from gx. Since G is countable we can number the possible states 1,2, and represent the Markov chain by its matrix of transitionprobabilities, M(G, A, P), say (cf. [l] for terminology).