Faster Mixing for Triangulations via Transport Flows

2026-05-03

AI Summary

This work addresses the long-standing gap between the best-known upper bound of $\widetilde{O}(n^3)$ and the theoretical lower bound of $\Omega(n^{3/2})$ for the mixing time of the flip Markov chain on triangulations of a convex $(n+2)$-gon. By developing a novel analysis framework based on transport flows, combined with combinatorial decomposition and functional inequality techniques, the authors significantly refine the characterization of the chain’s convergence rate. This approach yields improved upper bounds of $\widetilde{O}(n^2)$ for both the relaxation time and the logarithmic Sobolev constant, thereby tightening the mixing time bound from $\widetilde{O}(n^3)$ to $\widetilde{O}(n^2)$. The result represents a substantial step toward resolving Aldous’s conjecture that the true mixing time is $\Theta(n^{3/2})$.

Abstract

We prove an $\widetilde O(n^2)$ bound for the \emph{relaxation time} and the \emph{log-Sobolev time} (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex $(n+2)$-gon, implying a mixing time of $\widetilde O(n^2)$. The previous state of the art for the mixing time of this chain, due to Eppstein and Frishberg, was $\widetilde O(n^3)$, while the best known lower bound on the mixing time, due to Molloy, Reed, and Steiger, is $Ω(n^{3/2})$. Our relaxation time bound makes significant progress towards Aldous' conjectured bound of $Θ(n^{3/2})$ for the relaxation time. We improve upon the analysis of Eppstein and Frishberg by further developing the framework of \emph{transport flows} introduced in the work of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.

Problem

• triangulations
• mixing time
• relaxation time
• log-Sobolev time
• Markov chains

Innovation

• transport flows
• triangulation flip chain
• relaxation time
• log-Sobolev inequality
• mixing time