In my answer to [this question](https://math.stackexchange.com/questions/640/why-are-differentiable-complex-functions-infinitely-differentiable/820#820) on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way to make this precise is the following. The finitary analogue of a real-differentiable function is just a function $f : \mathbb{Z} \to \mathbb{Z}$, whose finitary derivative is the first difference $f(z + 1) - f(z)$. The first difference can be arbitrarily nasty; in particular it does not have to grow in any kind of smooth way, which is a reasonable analogue of what happens for arbitrary real-differentiable functions.


The finitary analogue of (the real part of) a complex-differentiable function, on the other hand, is a discrete harmonic function $f : \mathbb{Z}^2 \to \mathbb{Z}$ e.g. one which satisfies 


$$\frac{f(x+1, y) + f(x-1, y) + f(x, y+1) + f(x, y-1)}{4} = f(x, y).$$


Such functions are clearly more constrained than arbitrary functions $\mathbb{Z} \to \mathbb{Z}$. Is there a reasonable finitary statement about the properties of such functions which is analogous to the statement that harmonic functions are smooth?


(The best case would be if one could deduce the infinitary statement from the finitary one, but maybe this is difficult. Also, I can't think of good tags for this question, so feel free to retag.)



What is a reasonable finitary analogue of the statement that harmonic functions are smooth?
There are quite a few properties shared by discrete and continuous harmonic functions. They also generalize in various forms to graphs. I don't know if the analogy is so close that there is a unique concept of smoothness for the discrete case, but see the last item for one point of contact.


1. Poisson formula: $f(p)$ is a weighted average of the values of $f$ on a "curve" of points surrounding $p$ (that is, a minimal set of points disconnecting it from infinity). As in the continuous case the weighting is the distribution of hitting probabilities, $h\_p(x,y)$ = probability that a random walk started at $p$ first hits the curve at $(x,y)$.
2. Maximum principle. Because values in the interior are weighted averages of values on the boundary.
3. Growth constraints. A bounded discrete-harmonic function is constant. More generally there is a discrete analogue of the theorem that if a harmonic $f(x,y)$ has polynomial-bounded growth, $|f(x,y)| < C(|x|+|y|+1)^n$ then $f(x,y)$ is a (harmonic) polynomial in $x$ and $y$, of degree at most $n$.
4. Harnack inequalities. It is these that are a form of smoothness. They say that if $a$ and $b$ are nearby points far away from the boundary curve used in the Poisson formula, $f(a) - f(b)$ is small (relative to $|f|$ on the boundary). There are many expressions of this principle, see for example papers of Fan Chung and her collaborators ( www.math.ucsd.edu/~fan ) for inequalities in the case of graphs with a vertex-transitive symmetry group. By phrasing the theory in terms of eigenfunctions of the Laplacian one can give the discrete and continuous Harnack inequalities a very parallel form. But in the discrete case they also seem to play a more basic role of showing that smoothness is present in some form, which in the continuous case is automatic because $f$ is assumed to have at least two derivatives.


