TY - JOUR
T1 - Typical representatives of free homotopy classes in multi-punctured plane
AU - Arnold, Maxim
AU - Baryshnikov, Yuliy
AU - Mileyko, Yuriy
PY - 2019/9/1
Y1 - 2019/9/1
N2 - 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.
AB - 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.
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U2 - 10.1142/S1793525319500262
DO - 10.1142/S1793525319500262
M3 - Article
AN - SCOPUS:85038396101
VL - 11
SP - 623
EP - 659
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
SN - 1793-5253
IS - 3
ER -