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

2024/09/13 by Federico Ardila–Mantilla, Ardila-Mantilla, Federico, Naya Banerjee +3
Chemistry · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Chemistry and Stereochemistry Studies #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2409.09133

openalex publication_date 2024/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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