Probability of a topologically non-trivial random walk on a finte torus - MathOverflow

Probability of a topologically non-trivial random walk on a finte torus

2012-05-23T19:00:05Z

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

Answer by Robert Israel for Probability of a topologically non-trivial random walk on a finte torus

2012-05-23T19:22:11Z

Say the walk starts at $0$. The lattice in the torus is the image of ${\mathbb Z}^n$ under a quotient map, and your random walk on the torus is the image of a random walk on ${\mathbb Z}^n$. The walk on the torus gives a nontrivial loop iff the walk on ${\mathbb Z}^n$ hits some member of $L{\mathbb Z}^n$ other than $0$ before returning to $0$. If $n=2$, because random walk on ${\mathbb Z}^2$ is recurrent, the probability of a nontrivial loop goes to $0$ as $L \to \infty$. If $n \ge 3$, random walk on ${\mathbb Z}^n$ is transient; if this walk never returns to $0$ its first hit on $L {\mathbb Z}^n$ must not be at $0$, so the probability of a nontrivial loop has a nonzero lower bound.