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3 added 335 characters in body

As per Noah's answer and JBL's comment this was false as stated. However, I think the following reformulation is interesting.

As before we consider a random walk on $\mathbb{Z}^2$ where a particle either stays at its vertex or moves to a neighbor with probability 1/5. We start the process with a particle at the origin. For $x \in \mathbb{Z}^2$ we let $p_n(x)$ denote the probability that we find the particle at $x$ after $n$ iterations. Let $|\cdot|$ denote the Euclidean distance of two points in $\mathbb{Z}^2$ via the standard embedding of $\mathbb{Z}^2 \subset \mathbb{R}^2$.

Now for the reformulated question: For each $n$, let $C_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| \leq |y| \Rightarrow p_n(x) \geq p_n(y)$

Does $\lim_{n\to\infty} C_n = \infty$? If so, how fast does this diverge?

EDIT: As per George Lowther's comment, I now find it quite probable that $\lim\inf_{n\to\infty} C_n \leq 5$ if not $C_n = 5$ for all large $n$.

A natural attempt to salvage the question is the following: For each $n$, let $\tilde{C}_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| < |y| \Rightarrow p_n(x) > p_n(y)$

Again we ask if $\lim_{n\to\infty} \tilde{C}_n = \infty$ and if so, how fast this diverges.

2 added 147 characters in body

As per Noah's answer and JBL's comment this was false as stated. However, I think the following reformulation is interesting.

As before we consider a random walk on $\mathbb{Z}^2$ where a particle either stays at its vertex or moves to a neighbor with probability 1/5. We start the process with a particle at the origin. For $x \in \mathbb{Z}^2$ we let $p_n(x)$ denote the probability that we find the particle at $x$ after $n$ iterations. Let $|\cdot|$ denote the Euclidean distance of two points in $\mathbb{Z}^2$ via the standard embedding of $\mathbb{Z}^2 \subset \mathbb{R}^2$.

Now for the reformulated question: For each $n$, let $C_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| \leq |y| \Rightarrow p_n(x) \geq p_n(y)$

Does $\lim_{n\to\infty} C_n = \infty$? If so, how fast does this diverge?

EDIT: As per George Lowther's comment, I now find it quite probable that $\lim\inf_{n\to\infty} C_n \leq 5$ if not $C_n = 5$ for all large $n$.

1

# Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II

As per Noah's answer and JBL's comment this was false as stated. However, I think the following reformulation is interesting.

As before we consider a random walk on $\mathbb{Z}^2$ where a particle either stays at its vertex or moves to a neighbor with probability 1/5. We start the process with a particle at the origin. For $x \in \mathbb{Z}^2$ we let $p_n(x)$ denote the probability that we find the particle at $x$ after $n$ iterations. Let $|\cdot|$ denote the Euclidean distance of two points in $\mathbb{Z}^2$ via the standard embedding of $\mathbb{Z}^2 \subset \mathbb{R}^2$.

Now for the reformulated question: For each $n$, let $C_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| \leq |y| \Rightarrow p_n(x) \geq p_n(y)$

Does $\lim_{n\to\infty} C_n = \infty$? If so, how fast does this diverge?