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 welldefined. 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 $\alpha :{V}^{\prime}\to \mathbb{R}$ 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
$$f(v)=\mathbb{E}\alpha ({X}_{v})$$ is a harmonic function except on ${V}^{\prime}$.
The first example is a special case of this one, taking ${V}^{\prime}=\{a,z\}$ and $\alpha (a)=1,\alpha (z)=0$.
Title  example of harmonic functions on graphs 

Canonical name  ExampleOfHarmonicFunctionsOnGraphs 
Date of creation  20130322 12:45:53 
Last modified on  20130322 12:45:53 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  5 
Author  mathcam (2727) 
Entry type  Example 
Classification  msc 30F15 
Classification  msc 31C05 
Classification  msc 31B05 
Classification  msc 31A05 