Random walks on trees

The classical gambler's ruin problem, i.e., a random walk along a line may be viewed graph theoretically as a random walk along a path with the endpoints as absorbing states. This paper is an investigation of the natural generalization of this problem to that of a particle walking randomly on a tree with the endpoints as absorbing barriers. Expressions in terms of the graph structure are obtained from the probability of absorption at an endpoint e in a walk originating from a vertex v, as well as for the expected length of the walk.