Mixing on Generalized Associahedra

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 $B$.

19 pages, 6 figures