Random walks on graphs

Let G = (V,E) be a connected, undirected graph. We shall consider the following model of a random walk on G. The walk starts out at time 0 in some initial distribution on the vertex set V ; the initial condition may be deterministic, which corresponds to the distribution assigning probability 1 to a single vertex. The random walk is constructed as follows. There are independent unit rate Poisson processes associated with each edge of the graph. If the random walk is at some vertex v ∈ V at time s, its next jump occurs at the first time t > s at which there is an increment of the Poisson process on some edge (v, w) incident on V . If this increment occurs on edge (v, u), then the random walk moves to vertex u at time t. (The process is well defined because the probability of the next increment happening at exactly the same time on two different edges is zero.) Equivalently, if the random walk is at vertex v, it samples independent Exp(1) random variables for each edge (v, w) incident on v, and moves along the edge with the minimum value of these random variables; the time spent at v is equal to this minimum value. From this description, and the fact that the minimum of independent exponential random variables is exponential with the sum of their rates, it is clear that the time spent at a vertex v on each visit has an Exp(dv) distribution, where dv denotes the degree of the vertex v. Moreover, the next vertex to be visited after v is chosen uniformly at random from the neighbours of v.