Hello

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