Typical representatives of free homotopy classes in multi-punctured plane

Maxim Arnold, Yuliy Baryshnikov, Yuriy Mileyko

Research output: Contribution to journalArticlepeer-review


We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial 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 method for approximating shortest loops.

Original languageEnglish (US)
Pages (from-to)623-659
Number of pages37
JournalJournal of Topology and Analysis
Issue number3
StatePublished - Sep 1 2019


  • Mogulskii's theorem
  • Shortest loop

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology


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