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

Maxim Arnold; Yuliy Baryshnikov; Yuriy Mileyko

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arXiv · 1601.04224

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

Maxim Arnold·Yuliy Baryshnikov·Yuriy Mileyko

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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.

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BibTeX RIS

Maxim Arnold, Yuliy Baryshnikov, Yuriy Mileyko. 2016-01-16. Typical representatives of free homotopy classes in a multi-punctured plane. https://arxiv.org/abs/1601.04224

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