Take the 2-minute tour ×  I've casually proved, as application
of some ideas that I am developing, a result that might be of
interest in itself. I am completely new in this field and then I
would like to ask your help to understand: 1) might it be of
interest? 2) Is it trivial, in the sense that it can be proved
directly? 3) is it well-known?  Let me fix a planar and regular
setting, even if I could state the result in more general
settings: let $n\geq1$ be a fixed integer and
$P=[-n,n]^2\subseteq\mathbb Z^2$. Let me fix the following
notation: given $(x,y)\in P$, I will denote by $A(x,y)$ the set
formed by the following at most five points:
$(x-1,y),(x,y),(x+1,y),(x,y-1),(x,y+1)$, where at most means
that if one of those points does not belong to $P$, then I will
not consider it.  Now, the situation is the following: for any
point $p\in P$, let $\gamma_p$ a walk starting on $p$ and ending
on $p^+$. I suppose that: if a walk hits the boundary, then it
ends. In particular, if $p\in\partial P$, then $p^+=p$.  D