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

TL;DR

This paper demonstrates that in a multi-punctured plane, random loops within a free homotopy class tend to be close to the shortest possible loops, using an extended Mogulskii's theorem and MCMC sampling.

Contribution

It extends Mogulskii's theorem to closed paths and introduces a practical MCMC method for sampling minimal-length loops in free homotopy classes.

Findings

Probability measure concentrates around shortest loops

Extension of Mogulskii's theorem to closed paths

MCMC sampling effectively approximates shortest loops

Abstract

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