Harmonic functions and ends of graphs

Source abstract

In this note we discuss extensions of results [ 5 ], where transient random walks are considered, whose transition matrix is compatible with a tree-structure of the underlying discrete state space. Notation is generally as in [ 5 ], with the exception that instead of a tree T we consider an arbitrary graph Γ, which is locally finite, connected, and whose vertex set S is infinite. The edge set E is unoriented, there are no multiple edges. We consider a transition operator P , given by a stochastic matrix ( p ( u , v )) u , v ∈ S , which gives rise to a transient Markov chain (“random walk”), related to the graph structure by: (i) if p ( u, v )>0 then [ u, v ] ∈ E , (ii) there is an M >0 such that p ( k ) ( u, v )>0 for some k = k ( u, v )≦ M whenever [ u, v ]∈ E (“uniform irreducibility”).