Question

Consider a simple random walk on the lattice $\mathbb Z^2$ starting at the origin $(0,0)$ where in each step, one of the four adjacent vertices in chosen uniformly at random, i.e. with probability $1/4$. I want to know the distribution of the time that it takes to hit a certain vertex $(x,y)$. I have already done quite some research on related work but so far I was not able to come up with a paper that gives an answer to my question. I would appreciate any hint towards a paper that covers this problem or a hint on how I can calculate it myself.

Answer 1

Oops, too long for a comment:

The expectation of the time until you reach a given point other than the origin is infinite already in one dimension. To see this, let $T$ be the expected time until you get from 0 to 1 (in the obvious 1-dimensional setting). The first step is either to the left or to the right, and if you go to $-1$, the expected remaining time is going to be $2T$ since you have to get back to the origin and then to 1. Consequently $T = 1 + 1/2\cdot (2T)$, from which we see that finite $T$ leads to contradiction. In higher dimensions it becomes even worse. Presumably you want to modify your question and ask about something like the distribution of the time (in dimensions 1 and 2 you actually get there!) or about the expected time to get from one point to another in a finite box.