Mixing on Generalized Associahedra

TL;DR

This paper extends rapid mixing time results for simple random walks from the associahedron to type-B and type-D associahedra, providing bounds on mixing times and expansion properties.

Contribution

It adapts existing techniques to new associahedron types, establishing mixing time bounds and expansion properties for type-B and type-D associahedra.

Findings

Type-B associahedron mixing time: O(n^3 log^3 n)

Type-D associahedron mixing time: O(n^{13} log^2 n)

Expansion bounds are tight up to logarithmic factors in type B

Abstract

Eppstein and Frishberg recently proved that the mixing time for the simple random walk on the $1$-skeleton of the associahedron is $O(n^3\log^3 n)$. We obtain similar rapid mixing results for the simple random walks on the $1$-skeleta of the type-$B$ and type-$D$ associahedra. We adapt Eppstein and Frishberg's technique to obtain the same bound of $O(n^3\log^3 n)$ in type $B$ and a bound of $O(n^{13} \log^2 n)$ in type $D$; in the process, we establish an expansion bound that is tight up to logarithmic factors in type $B$.