Random walk to visible lattice points

Question

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.

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

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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 $34$%?

Answer 1

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 $r$.

Answer 2

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.