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 ${𝒯}_3=(V_3,E_3)$ be a 3-regular tree. Assign “levels” to the vertices of ${𝒯}_3$ as follows: Fix a vertex $o\in V_3$, and let $\pi$ be a branch of ${𝒯}_3$ (an infinite simple path) from $o$. For every vertex $v\in V_3$ of ${𝒯}_3$ there exists a unique shortest path from $v$ to a vertex of $\pi$; let $\mathrm{\ell }(v)=\left|\pi \right|$ be the length of this path.

Now define $f(v)=2^{-\mathrm{\ell }(v)}>0$. Without loss of generality, note that the three neighbours $u_1,u_2,u_3$ of $v$ satisfy $\mathrm{\ell }(u_1)=\mathrm{\ell }(v)-1$ (“$u_1$ is the parent of $v$”), $\mathrm{\ell }(u_2)=\mathrm{\ell }(u_3)=\mathrm{\ell }(v)+1$ (“$u_2,u_3$ are the siblings of $v$”). And indeed,$\frac{1}{3}\left(2^{\mathrm{\ell }(v)-1}+2^{\mathrm{\ell }(v)+1}+2^{\mathrm{\ell }(v)+1}\right)=2^{\mathrm{\ell }(v)}$.

So $f$ is a positive nonconstant harmonic function on ${𝒯}_3$.