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.

中文翻译：

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多穿孔平面上自由同伦类的典型代表

我们表明，在多穿孔平面中的非平凡自由同伦类中，在一组特定分段线性循环上支持的统一概率度量绝大多数集中在最小长度的循环周围。我们的方法基于将 Mogulskii 定理扩展到封闭路径，这是独立兴趣的有用结果。此外，我们表明可以使用标准马尔可夫链蒙特卡罗技术对上述度量进行采样，从而提供一种近似最短循环的简单方法。