Given $n\in \mathbb N$, describe the set of all the discrete harmonic functions $f:\mathbb Z^n \to \mathbb R_+$.
A function $f:\mathbb Z^n\to \mathbb R$ is harmonic iff $$ 2nf(x) = \sum_{i=1}^n [f(x-e_i)+f(x+e_i)], $$ for all $x\in \mathbb Z^n$, where $(e_1,\ldots,e_n)$ is the canonical basis of $\mathbb Z^n$. Note that $f$ is discrete harmonic iff $f$ is a martingale w.r.t. a symmetric random walk on $\mathbb Z^n$.
It was argued that for $n=2$ all the discrete harmonic functions are constant. However, that assumes that $f$ is bounded from both below and above. Is the lower bound sufficient to eliminate all the non-constant functions?
For $n=1$ it is easy to show that $f$ has to be constant as all the discrete harmonic functions on $\mathbb Z$ need to have constant growth, so if $f$ was not constant it would have to be unbounded from both below and above. However, for $n=2$, the set of discrete harmonic functions on $\mathbb Z^2$ is much more rich.
Similar: A discussion of the related question for (not discrete) harmonic functions.
