Bounded harmonic function theorem on a recurrent graph ($h(v)=h(w)(v,w\in V)$)

Every bounded harmonic function on a graph is constant if the connected locally finite graph is recurrent. Along its random walk on a graph, the harmonic values form a bounded martingale and converge almost surely. Recurrence visits each vertex infinitely often, making every vertex value a subsequential limit of the same convergent sequence. Thus every value agrees.