 
Summary: Math. Proc. Camb. Phil. Soc. (1989), 106. 179 1 7 9
Printed in Great Britain
Hitting times for random walks on vertextransitive graphs
BY DAVID ALDOUS
Department of Statistics, University of California, Berkeley CA 94720, U.S.A.
(Received 4 July 1988)
Abstract
For random walks on finite graphs, we record some equalities, inequalities and
limit theorems (as the size of graph tends to infinity) which hold for vertextransitive
graphs but not for general regular graphs. The main result is a sharp condition for
asymptotic exponentiality of the hitting time to a single vertex. Another result is a
lower bound for the coefficient of variation of hitting times. Proofs exploit the
complete monotonicity properties of the associated continuoustime walk.
1. Introduction
Random walks on graphs have been studied in a wide variety of contexts. On
highlysymmetric (e.g. distancetransitive) graphs it is feasible to attempt analytic
calculations of wstep transition probabilities and exact hitting time distributions:
see [10, 16, 18]. At the other extreme, for general graphs there are various general
bounds known [5, 1, 4] and in the more general setting of reversible Markov chains
there are techniques for obtaining longrange estimates [20].
