Hello

Suposse we have a particle in the plane at the origin $\displaystyle (0,0).$

It moves randomly on the integer lattice $\displaystyle Z^2$ to any of the adjacent vertex with equal probability $\displaystyle 1/4$.

The question is what's the probabiliy of reach a fixed point $\displaystyle (x,y)$ before return to the origin?

The analog problem in one dimension is easy. Such probability is:

$\displaystyle

\dfrac{1}{2|x|}

$

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