Suppose a sphere is partitioned by a latitude-longitude grid, with grid quadrilaterals $\Delta \times \Delta$. All grid nodes have degree $4$, while the North & South poles have degree $2 \pi / \Delta$, with incident grid triangles.

Let a random walk from any grid node $p$ take one step along each an incident arc, each with equal probability. Here is a $1000$-step random walk starting from the North pole, with $\Delta = 10^\circ$.

In this example, the north pole is visited $20$ times. My question is:

. What is the distribution of visits to the grid nodes (as a function of $n$, the number of steps, and $\Delta$)?Q

. Specifically, how many visits to each pole can be expected (starting anywhere)?Qa

. Are the nodes closer to the poles (excluding the poles themselves) visited more frequently than those near the equator?Qb

Clearly the poles are visited frequently.
In a trial of $n{=}10000$ steps, $\Delta = 10^\circ$, the north pole was visited $195$ times.
Although I expected the answer to ** Qb** to be

*Yes*, I am not observing clear evidence of such a bias.