We prove an $\widetilde O(n^2)$ bound for the relaxation time and the 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 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：我们证明了在凸$(n+2)$边形上经典三角剖分翻转链的松弛时间和对数索伯列夫时间（即对数索伯列夫常数的倒数）的上界为$\widetilde O(n^2)$，这意味着混合时间的上界为$\widetilde O(n^2)$。此前，由Eppstein和Frishberg得到的该链混合时间的最佳上界为$\widetilde O(n^3)$，而Molloy、Reed和Steiger给出的已知最佳下界为$Ω(n^{3/2})$。我们的松弛时间界为Aldous推测的松弛时间界$Θ(n^{3/2})$取得了重要进展。我们通过进一步发展Chen等人工作中引入的传输流框架，改进了Eppstein和Frishberg的分析方法。在此视角下，我们的结果可视为利用组合分解获得马尔可夫链泛函不等式的一种更高效方式。我们希望本文的思想未来能获得其他应用。