Hi: this question is regarding the topological properties of random walks on a finite torus.

Consider an unbiased random walk on finite square lattice on a torus of linear dimension $L$. Place a trap at the origin, such that the random walk ends as soon as the walker lands in the trap.

If the walker begins at the origin, what is the probability that the path of the random walker will form a topologically trivial (contractible) loop on the torus after it has returned to the origin for the first time and is trapped?

More generally, I am interested in the probabilities the path having an arbitrary topological winding number at the first-return of the walker, and how this depends (or does not depend) on $L$.

thanks,

Chris