On Random Graph Homomorphisms into Z

Given a bipartite connected finite graph G=(V, E) and a vertex v0∈V, we consider a uniform probability measure on the set of graph homomorphisms f: V→Z satisfying f(v0)=0. This measure can be viewed as a G-indexed random walk on Z, generalizing both the usual time-indexed random walk and tree-indexed random walk. Several general inequalities for the G-indexed random walk are derived, including an upper bound on fluctuations implying that the distance d(f(u), f(v)) between f(u) and f(v) is stochastically dominated by the distance to 0 of a simple random walk on Z having run for d(u, v) steps. Various special cases are studied. For instance, when G is an n-level regular tree with all vertices on the last level wired to an additional single vertex, we show that the expected range of the walk is O(log n). This result can also be rephrased as a statement about conditional branching random walk. To prove it, a power-type Pascal triangle is introduced and exploited.