Bounded harmonic function theorem on a recurrent graph
= Bounded harmonic function theorem on a recurrent graph
{title2=$h(v)=h(w)\quad(v,w\in V)$}
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.