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.

翻译：我们证明，在凸$(n+2)$边形上的经典三角剖分翻转链中，\emph{松弛时间}和\emph{对数索伯列夫时间}（对数索伯列夫常数的倒数）具有$\widetilde O(n^2)$的上界，这意味着混合时间为$\widetilde O(n^2)$。此前，由Eppstein和Frishberg给出的该链混合时间的最佳上界为$\widetilde O(n^3)$，而由Molloy、Reed和Steiger给出的混合时间最佳下界为$Ω(n^{3/2})$。我们的松弛时间上界显著推进了Aldous关于松弛时间为$Θ(n^{3/2})$的猜想。通过进一步发展Chen等人工作中引入的\emph{传输流}框架，我们改进了Eppstein和Frishberg的分析。在此视角下，我们的结果可视为利用组合分解来获取马尔可夫链泛函不等式的一种更高效方法。我们希望未来这些思想能应用于其他领域。