Two-dimensional random walk

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

@el_manco - Hi - This has been troubling me since you posted this.

Infact I am not even to able to prove your point about 1-D random walk.

Can anyone help me with the 1-D problem?

And any insights about 2-D problem