# example of harmonic functions on graphs

1. 1.

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

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

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. 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

 $f(v)=\operatorname{\mathbb{E}}\alpha(X_{v})$

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 ExampleOfHarmonicFunctionsOnGraphs 2013-03-22 12:45:53 2013-03-22 12:45:53 mathcam (2727) mathcam (2727) 5 mathcam (2727) Example msc 30F15 msc 31C05 msc 31B05 msc 31A05