Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start the random walk with one particle at the origin. For each $n \geq 1$ and $x \in \mathbb{Z}^2$ let $p\_n(x)$ be the probability of finding the particle at $x$ after $n$ iterations. For two points $x,y \in \mathbb{Z}^2$ let $|\cdot|$ denote the Euclidean distance of $x$ and $y$ via the standard embedding $\mathbb{Z}^2 \subset \mathbb{R}^2$. 


For what $n$ is it true that $|x| \leq |y| \Rightarrow p\_n(x) \geq p\_n(y)$? What kind of techniques are available to prove statements like this? Barring arithmetic mistakes I have verified this up to n=6 via explicit computation.


Please forgive me if this is actually a trivial question (I know very little about random walks). I would also be very happy with suggested approaches or references.


A Little Motivation/Another Problem: Suppose we list the elements of $\mathbb{Z}^2$ is ascending order by Euclidean distance from the origin, $z\_1 \leq z\_2 \leq \cdots$, and then set $D\_n = \cup\_{i=1}^n z\_i$. For various reasons I have been dealing with these $D\_n$ and would like to consider analogues in other groups. Hence I would very much like to have a "$\mathbb{Z}^2$-intrinsic" characterization of these $\{D\_n\}$, i.e. it would be nice to have a characterization of $D\_n$ that only used group or graph theoretic statements about $\mathbb{Z}^2$. Most importantly I do not want to mention the specific embedding of $\mathbb{Z}^2$ into $\mathbb{R}^2$.


Note: The $D\_n$ are not exactly well defined since there are choices involved in the list $z\_1 \leq z\_2 \leq \cdots $. So I am actually interested in characterizing them up to the forced ambiguity. 



Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
As written the statement is false for $n=3$: note that $p\_3(2,2) = 0$ but $p\_3(3,0) > 0$, while $|(2,2)| < |(3,0)|$. Similar counterexamples exist for all $n\geq 5$. So for larger $n$ you would at least need some extra condition about $L^1$ norms to guarantee that you can't have $|x|<|y|$ with $p\_n(x)=0$ and $p\_n(y)>0$. I would guess that this would still be too weak, however.


