Liouville property of graphs

Summary

A graph has the Liouville property if every bounded discrete harmonic function on it is constant. The name comes from Liouville's theorem, whose harmonic form says that bounded harmonic functions on $\R^n$ are constant. The lattices $\Z^d$ have the property, while regular trees of degree at least $3$ do not. For Cayley graphs of groups it says that the Poisson boundary of the simple random walk is trivial.