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

TL;DR

This paper demonstrates that certain Markov chains on monotone paths in a strip exhibit slow mixing, utilizing CAT(0) cubical complexes to identify bottlenecks, and provides formulas for counting these paths with growth rate analysis.

Contribution

It introduces a novel application of CAT(0) cubical complexes to analyze mixing times of Markov chains on monotone paths and derives a general formula for path enumeration.

Findings

Markov chains on monotone paths mix slowly.

A formula for counting monotone paths c_m(n) is established.

The exponential growth rate of c_m(n) is computed for any m.

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.