Summary: Math. Proc. Camb. Phil. Soc. (1989), 106. 179 1 7 9
Printed in Great Britain
Hitting times for random walks on vertex-transitive graphs
BY DAVID ALDOUS
Department of Statistics, University of California, Berkeley CA 94720, U.S.A.
(Received 4 July 1988)
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 vertex-transitive
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 continuous-time walk.
Random walks on graphs have been studied in a wide variety of contexts. On
highly-symmetric (e.g. distance-transitive) graphs it is feasible to attempt analytic
calculations of w-step 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 long-range estimates .