We prove that the well-studied triangulation flip walk on a convex point set mixes in time O(n^3 log^3 n), the first progress since McShine and Tetali's O(n^5 log n) bound in 1997. In the process we give lower and upper bounds of respectively Omega(1/(sqrt n log n)) and O(1/sqrt n) -- asymptotically tight up to an O(log n) factor -- for the expansion of the associahedron graph K_n. The upper bound recovers Molloy, Reed, and Steiger's Omega(n^{3/2}) bound on the mixing time of the walk. To obtain these results, we introduce a framework consisting of a set of sufficient conditions under which a given Markov chain mixes rapidly. This framework is a purely combinatorial analogue that in some circumstances gives better results than the projection-restriction technique of Jerrum, Son, Tetali, and Vigoda. In particular, in addition to the result for triangulations, we show quasipolynomial mixing for the k-angulation flip walk on a convex point set, for fixed k >= 4.

翻译：我们证明，在凸点集上广泛研究的三角剖分翻转游走可在时间O(n^3 log^3 n)内实现混合，这是自1997年McShine和Tetali的O(n^5 log n)界以来的首次进展。在此过程中，我们给出了结合多面体图K_n的扩张下界和上界，分别为Omega(1/(√n log n))和O(1/√n)——渐近紧确至多一个O(log n)因子。上界恢复了Molloy、Reed和Steiger关于游走混合时间的Omega(n^{3/2})界。为获得这些结果，我们引入一个由一组充分条件构成的框架，在该框架下给定的马尔可夫链能够快速混合。该框架是纯组合学的类比，在某些情况下比Jerrum、Son、Tetali和Vigoda的投影限制技术效果更优。具体而言，除了三角剖分的结果外，我们还证明了对于固定的k ≥ 4，凸点集上k-角剖分翻转游走的拟多项式混合时间。