# a harmonic function on a graph which is bounded below and nonconstant

There exists no harmonic function on all of the $d$-dimensional grid $\mathbb{Z}^{d}$ which is bounded below and nonconstant. This categorises a particular property of the grid; below we see that other graphs can admit such harmonic functions.

Let $\mathcal{T}_{3}=(V_{3},E_{3})$ be a 3-regular tree. Assign “levels” to the vertices of $\mathcal{T}_{3}$ as follows: Fix a vertex $o\in V_{3}$, and let $\pi$ be a branch of $\mathcal{T}_{3}$ (an infinite simple path) from $o$. For every vertex $v\in V_{3}$ of $\mathcal{T}_{3}$ there exists a unique shortest path from $v$ to a vertex of $\pi$; let $\ell(v)=\left|\pi\right|$ be the length of this path.

Now define $f(v)=2^{-\ell(v)}>0$. Without loss of generality, note that the three neighbours $u_{1},u_{2},u_{3}$ of $v$ satisfy $\ell(u_{1})=\ell(v)-1$ (“$u_{1}$ is the parent of $v$”), $\ell(u_{2})=\ell(u_{3})=\ell(v)+1$ (“$u_{2},u_{3}$ are the siblings of $v$”). And indeed, $\frac{1}{3}\left(2^{\ell(v)-1}+2^{\ell(v)+1}+2^{\ell(v)+1}\right)=2^{\ell(v)}$.

So $f$ is a positive nonconstant harmonic function on $\mathcal{T}_{3}$.

Title a harmonic function on a graph which is bounded below and nonconstant AHarmonicFunctionOnAGraphWhichIsBoundedBelowAndNonconstant 2013-03-22 12:44:26 2013-03-22 12:44:26 drini (3) drini (3) 6 drini (3) Example msc 30F15 msc 31C05 msc 31B05 msc 31A05