example of harmonic functions on graphs

  1. 1.

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

    f(v)={simple random walk from v hits a before z}

    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 walkMathworldPlanetmath on G is recurrent”.

  2. 2.

    Let G=(V,E) be a graph, and let VV. Let α:V be some boundary conditionMathworldPlanetmath. For uV, define a random variableMathworldPlanetmath Xu to be the first vertex of V that simple random walk from u hits. The function


    is a harmonic function except on V.

    The first example is a special case of this one, taking V={a,z} and α(a)=1,α(z)=0.

Title example of harmonic functions on graphs
Canonical name ExampleOfHarmonicFunctionsOnGraphs
Date of creation 2013-03-22 12:45:53
Last modified on 2013-03-22 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