Spiral lattice random walk

Question

Execute a random walk from the origin on the integer lattice, but bias the four compass-direction probabilities from $\frac{1}{4}$ each to prefer to step in a spiraling direction. Calling the four step vectors $c_0,c_1,c_2,c_3$, with $c_i=(\cos (i \, \pi/2), \sin (i \, \pi/2))$, adjust the probabilities as follows. Let $v$ be the vector from the origin to the last point on the path, and $n$ the unit normal to $v$, counterclockwise $90^\circ$ to $v$. Then select step $c_i$ with probability $\frac{1}{4} (1+ c_i \cdot n)$.

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          ^{$\theta=60^\circ$. $\frac{1}{4}(1+c_2 \cdot n) = (1+\sqrt{3}/2)/4 \approx 0.47$.}

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Unsurprisingly, the random walks spiral around the origin:

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^{$2000$-step random walks. Origin: green. Last point: red. Three examples, followed by five examples at reduced scale.}

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Q. Does Pólya's recurrence theorem hold for these walks? Do the walks return to the origin with probability $1$?

All but one of the above examples (the penultimate) returned to the origin, but usually rather quickly.

Answer 1

It seems to me that this random walk is recurrent. Denote $Y_n=\|X_n\|$, where $(X_n, n\geq 0)$ is your "spiral" walk. Then, as $x\to \infty$, my calculations imply that $$ \mathbb{E}(Y_{n+1}-Y_n\mid Y_n=x) = \frac{1}{4\|x\|} + O(\|x\|^{-2}), $$ and $$ \mathbb{E}((Y_{n+1}-Y_n)^2\mid Y_n=x) = \frac{1}{2} + O(\|x\|^{-1}). $$ Then, (null) recurrence follows from Theorem 3.5.2 of [Menshikov, Popov, Wade, "Non-homogeneous random walks", C.U.P.-2017, http://www.ime.unicamp.br/~popov/book_lyapunov.pdf ].

Answer 2

Here is the theorem cited in Serguei Popov's answer:

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and here are the three assumptions:

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Answer 3

Forgive this simple-minded approach, but it may be instructive when made rigorous by someone who knows what they are doing.

I think Anthony Quas's suggestion of looking at radius good, but I do not understand his use of the word 'recurrent'. The probability of going east-west is 1/2, as is going north-south. However, most of the time, the probability for increasing the distance from the origin is greater than 1/2. This is because one of (say) east west is biased toward increasing distance, while the other (say) is half increasing and half decreasing. So for near the origin, there is a chance of returning, but (as Martin Hairer suggests) once you get far enough away, the dynamic seems transient.

I turn the calculation of "far enough away" over to those with more experience. I suspect the probability of returning to within S distance of the origin given one is at distance R in this dynamic is exponential in S-R (I assume S less than R).

Gerhard "Wonders About Inward Spiral Dynamics" Paseman, 2017.06.23.