Probability of a topologically non-trivial random walk on a finte torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:53:49Z http://mathoverflow.net/feeds/question/97782 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97782/probability-of-a-topologically-non-trivial-random-walk-on-a-finte-torus Probability of a topologically non-trivial random walk on a finte torus Chris 2012-05-23T19:00:05Z 2012-05-23T19:22:11Z <p>Hi: this question is regarding the topological properties of random walks on a finite torus.</p> <p>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. </p> <p>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?</p> <p>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 $L$.</p> <p>thanks,</p> <p>Chris </p> http://mathoverflow.net/questions/97782/probability-of-a-topologically-non-trivial-random-walk-on-a-finte-torus/97783#97783 Answer by Robert Israel for Probability of a topologically non-trivial random walk on a finte torus Robert Israel 2012-05-23T19:22:11Z 2012-05-23T19:22:11Z <p>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.</p>