We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of $$n^{-5/4}$$ n - 5 / 4 and a lower bound of $$n^{-11/2}$$ n - 11 / 2 . In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees—or, equivalently, on Dyck paths—and improve the previous lower bound for its spectral gap by Shor and Movassagh.

中文翻译：

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四边形边缘翻转的多项式混合时间

我们为有根四边形上的随机边缘翻转的混合时间建立了第一个多项式上限：我们证明了具有 n 个面的四边形上的边缘翻转马尔可夫链的光谱间隙承认，直到常数，上限为 $$n^ {-5/4}$$ n - 5 / 4 和 $$n^{-11/2}$$ n - 11 / 2 的下限。为了获得下界，我们还考虑了梧桐树上的一个非常自然的马尔可夫链——或者等效地，在 Dyck 路径上——并改进了 Shor 和 Movassagh 之前的光谱间隙下界。