If I had an integer lattice on a plane and a particle on a point on the lattice in the first quadrant, what is a good way to model the probability of the particle reaching either the x or y axis? The probability of the particle going up, down, left, right is always 1/4. And the particle always moves on each step.

I'm assuming I need to consider Markov chains and martingales, but any advice would be helpful!