Hello

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

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

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

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

$ \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

Suppose we have a particle in the plane at the origin $(0,0)$. It moves randomly on the integer lattice $Z^2$ to any of the adjacent vertexs with equal probability $1/4$. What's the probability of reaching a fixed point $(x,y)$ before returning to the origin?

The analogous problem in one dimension is easy. The probability is:

$ \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