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Markov chains, CAT(0) cube complexes, and enumeration: monotone paths in a strip mix slowly
Federico Ardila-Mantilla, Naya Banerjee, Coleson Weir · 2024-09-14 · via math.PR updates on arXiv.org

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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