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There is the following estimation (Duffin, Discrete potential theory, Theorem 5):

Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if $f$ is non-negative in this sphere, then for each neighbor $p'$ of $p$ we have $|f(p)-f(p')|\leq C f(p)/R$ for some absolute constant $C$.

It seems that the proof can be generalised for the integer lattices of all dimensions. Do you know any reference for that?

Also, Duffin uses Fourier transform and estimates for Green function. Probably, there is a simpler way to prove this statement...

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    $\begingroup$ Why don't you just repeat the proof for ordinary harmonic functions, using Harnack and Poisson's formula. All these things are available in the discrete setting. $\endgroup$
    – Alexandre Eremenko
    Commented May 14, 2015 at 23:59
  • $\begingroup$ I would rather ask about a reference. I am sure this fact is proven somewhere and it is much more easy to put a link than to write the proof in an article. $\endgroup$
    – Nikita Kalinin
    Commented May 22, 2015 at 14:40
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    $\begingroup$ The old book of F. Spitzer, Principles of random walk, almost surely contains this, but I do not have it beside me at the moment. $\endgroup$
    – Alexandre Eremenko
    Commented May 22, 2015 at 20:42
  • $\begingroup$ @ Alexandre Eremenko, I did not find it there. Nothing similar at all, no estimates for the derivative of a Green function or whatever. $\endgroup$
    – Nikita Kalinin
    Commented Jul 3, 2015 at 18:21

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