Mixing on Generalized Associahedra

TL;DR

The paper studies rapid mixing of simple random walks on the 1-skeleta of generalized associahedra of types $B$ and $D$. It adapts a multicommodity-flow framework to bound expansion and, through projection-graph techniques, derives explicit mixing-time bounds: $O(n^3 \log^3 n)$ for type $B$ (with a near-tight expansion $h(\mathfrak b_n)=\Omega(1/(\sqrt{n}\log n))$) and $O(n^{13} \log^2 n)$ for type $D$ (with $h(\mathfrak d_n)=\Omega(1/(n^5 \log n))$). The analysis leverages subgraph flows isomorphic to type $A$ associahedra, careful handling of overlapping classes, and refined concentration arguments to achieve the bounds. These results extend rapid mixing phenomena from type $A$ to the broader, combinatorially rich family of generalized associahedra, with implications for sampling in cluster algebras and related combinatorial structures.

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 $B$.

Figures (6)

• Figure 1: The graph $\mathfrak b_3$. Triangulations of the same color belong to the same class, meaning they have the same central diagonal.
• Figure 2: An illustration of part of the proof of \ref{['thm:typeBloose']} with $n=4$. Selecting an edge in $\mathcal{E}(D,D')$ is equivalent to choosing triangulations of $\overline R_1$ and $\overline R_2$.
• Figure 3: Two central diagonals $D$ and $D'$ of $P_{14}$ with $\Psi(D)\subseteq S$ and $\Psi(D')\subseteq \overline S$. The rectangle whose diagonals are $D$ and $D'$ breaks $P_{14}$ into two $4$-gons (blue) and two $5$-gons (green), so $d(D,D')=3$.
• Figure 4: The graph $\mathfrak d_3$. We have color-coded the centrally symmetric pairs of central chords.
• Figure 5: The pairs $ \boldsymbol{r}$ and $ \boldsymbol{r}$ of central chords in $P_{6}^\bullet$, where $\boldsymbol{r}$ is the pair of vertices marked with small circles.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: jsconductancesinclair_1992
• Lemma 2.2
• Corollary 2.3
• Lemma 2.4: eppstein2022improved
• Remark 3.1
• Proposition 3.2
• proof