Suposse we have a particle in the plane at the origin
It moves randomly on the integer lattice to any of the adjacent vertex with equal probability .
The question is what's the probabiliy of reach a fixed point before return to the origin?
The analog problem in one dimension is easy. Such probability is:
I have read some related articles working on finite graphs; but I am not be able to obtain the answer for my problem.
Thank you very much for your attention