马尔可夫链、CAT(0)立方体复合物和枚举：带状混合体中的单调路径缓慢变化

arXiv - MATH - Probability Pub Date : 2024-09-13 DOI:arxiv-2409.09133

Federico Ardila-Mantilla, Naya Banerjee, Coleson Weir

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引用次数: 0

摘要

我们证明了星状图中单调路径集合上的两条自然马尔可夫链会缓慢混合。为此，我们新颖地使用了非正曲（CAT(0)）立方复曲面理论，以检测许多具有混杂性的图中的小瓶颈。同时，我们给出了高度为 m 的带状图中长度为 n 的单调路径的数量 c_m(n)的计算公式。特别是，我们计算了任意 m 的 c_m(n)的指数增长常数，推广了威廉姆斯关于 m=2, 3 的结果。

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Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly

We prove that two natural Markov chains on the set of monotone paths in a strip mix slowly. To do so, we make novel use of the theory of non-positively curved (CAT(0)) cubical complexes to detect small bottlenecks in many graphs of combinatorial interest. Along the way, we give a formula for the number c_m(n) of monotone paths of length n in a strip of height m. In particular we compute the exponential growth constant of c_m(n) for arbitrary m, generalizing results of Williams for m=2, 3.

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arXiv - MATH - Probability

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