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Consider a random walk from the $\mathbb{Z}^2$ origin $(0,0)$ to visible (not blocked) lattice points $p$, with a parameter $r$ a given radius of a circle centered on $p$. With $p$ the previous point, the next point $p=p+(a,b)$ must have:

  • $a$ and $b$ co-prime. So $p$ can see $p+(a,b)$.
  • $a^2+b^2 \le r^2$. So $p$ lies in or on the radius-$r$ circle centered on $p$.
  • $(a,b)$ is chosen with probability proportional to its inverse distance squared, $1/(a^2+b^2)$. So, long steps are possible but increasingly rare with the length of the step.

Concerning this last point, see "Illumination from visible lattice points with inverse square intensity". The walk is something akin to a walk between (lattice) stars.

r10k100s4

Example: $r=10$, $n=100$ steps. Green is origin $(0,0)$. Last point $(-16, -7)$ red.

Q1. For a fixed $r$, is this walk recurrent, i.e., does it return to the origin with probability $1$? If instead transient, what is the probability of returning to the origin?

Q2. What is the behavior of the walk as $r \to \infty$?

Q3. What is the situation in $\mathbb{Z}^3$, where it is known that a standard random walk returns to the the origin with probability $34$%?

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  • $\begingroup$ or $Q2$, what specific questions behavior do you have in mind? Large-scale behavior (for a fixed $R$ and time going to infinity ) is universal, the walk converges to a Brownian motion whose variance is is governed by the second moment of the transition kernel. $\endgroup$
    – Kostya_I
    Commented Sep 17, 2023 at 18:01
  • $\begingroup$ @Kostya_I: You answered with convergence to Brownian motion. Thanks! $\endgroup$ Commented Sep 17, 2023 at 20:53

2 Answers 2

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The step distribution $\mu$ of your random walk is finitely supported and symmetric, so that all standard results are applicable. In particular, your random walk is recurrent for any value of the parameter $r$.

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  • $\begingroup$ Concerning dimension 3 finding an asymptotic (as $r\to\infty$) of the return probability might be an interesting question. The standard approach is to use harmonic analysis, but it's not immediately clearly, how friendly it is with the class of step distributions you consider. $\endgroup$
    – R W
    Commented Sep 18, 2023 at 19:08
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To give a specific reference, T1 in Section 8, Chapter II of Spitzer: Principles of random walks asserts that a 2D random walk is recurrent if $\sum_{x\in\mathbb{Z}^2}|x^2|P(0,x)<\infty$, where $P$ are transition probabilities. In dimension $3$ and higher, a random walk not supported on a 2D plane is always transient.

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