example of harmonic functions on graphs

1. Let $G=(V,E)$ be a connected finite graph, and let $a,z\in V$ be two of its vertices. The function

   $$f(v)=\mathbb{P}\left\{\text{simple random walk from }v\text{ hits }a\text{ before }z\right\}$$

   is a harmonic function except on $\{a,z\}$.

   Finiteness of $G$ is required only to ensure $f$ is well-defined. So we may replace “$G$ finite” with “simple random walk on $G$ is recurrent”.

2. Let $G=(V,E)$ be a graph, and let $V^{\prime }\subseteq V$. Let $\alpha :V^{\prime }\to ℝ$ be some boundary condition. For $u\in V$, define a random variable $X_u$ to be the first vertex of $V^{\prime }$ that simple random walk from $u$ hits. The function

   $$f(v)=𝔼\alpha (X_v)$$

   is a harmonic function except on $V^{\prime }$.

   The first example is a special case of this one, taking $V^{\prime }=\{a,z\}$ and $\alpha (a)=1,\alpha (z)=0$.