Execute a random walk from the origin on the integer lattice, but bias the four compass-direction probabilities from 14 each to prefer to step in a spiraling direction. Calling the four step vectors c0,c1,c2,c3, with ci=(cos(iπ/2),sin(iπ/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∘ to v. Then select step ci with probability 14(1+ci⋅n).
θ=60∘. 14(1+c2⋅n)=(1+√3/2)/4≈0.47.
Unsurprisingly, the random walks spiral around the origin:
2000-step random walks. Origin: green. Last point: red. Three examples, followed by five examples at reduced scale.
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.