Typical representatives of free homotopy classes in a multi-punctured plane

摘要

We show that a uniform probability measure supported on a specific set ofpiecewise linear loops in a non-trivial free homotopy class in amulti-punctured plane is overwhelmingly concentrated around loops of minimallengths. Our approach is based on extending Mogulskii's theorem to closedpaths, which is a useful result of independent interest. In addition, we showthat the above measure can be sampled using standard Markov Chain Monte Carlotechniques, thus providing a simple methods for approximating shortest loops.