---
tag: 7F5MZS55Y1
kind: definition
name: Liouville property of graphs
msc: [31C20, 05C81]
defines: Y1EGR8DF84
uses: [00C0BR7D7H, WRY0M7SW59, 062D9KXJB1, 04VZE3FZXF]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_function
---

## Definition

Let $G = (V, E)$ be a connected infinite [[graph|00C0BR7D7H]] in which every vertex has finitely many neighbours, with symmetric positive edge weights $c$.
The graph $G$ has the *Liouville property* if every bounded function $f \colon V \to \R$ that is [[harmonic|WRY0M7SW59]] on all of $V$ is constant.
It has the *strong Liouville property* if every positive function that is harmonic on all of $V$ is constant.
Without weights, the edge weights are taken equal to $1$, which corresponds to the simple [[random walk|04VZE3FZXF]].

## Remarks

Example: $\Z^d$ with unit weights has the Liouville property for every $d \ge 1$.
For $d = 1$ this is direct, since the harmonic functions on $\Z$ are $f(n) = an + b$, and such a function is bounded only when $a = 0$.
Non-example: the $3$-regular [[tree|062D9KXJB1]] $T_3$ fails it.
Fix an edge $\set{o, o'}$, and let $h(x)$ be the probability that the simple random walk from $x$ eventually stays in the component of $T_3$ minus this edge that contains $o'$.
The walk on $T_3$ is transient, so it crosses the edge finitely often, and $h$ is harmonic by conditioning on the first step.
Its values lie in $[0, 1]$, and it is not constant.
Indeed $h(x)$ tends to $1$ as $x$ moves away from the edge on the side of $o'$, and to $0$ on the side of $o$.
The strong Liouville property implies the Liouville property, since a bounded harmonic $f$ gives the positive harmonic function $f - \inf f + 1$.