论文标题
在平面图上旋转的混合时间的多项式上限
A polynomial upper bound for the mixing time of edge rotations on planar maps
论文作者
论文摘要
我们认为在所有根生根的平面地图上都具有自然的局部动态，并具有$ n $边缘，在某种意义上，它类似于“边缘翻转”马尔可夫链，这是在各种组合结构上已经考虑过的（$ n $ gon-gon-gon-gon-gon-gon-gon-gon-gon-godrangers and Quadranglastions of the Sphere等）。我们为平面图上的“边缘旋转”链的混合时间提供了第一个多项式上限：我们证明边缘旋转链的光谱间隙在下面以适当的恒定时间$ n^{ - 11/2} $界定。在此过程中，我们提供了一个新的新证明，即相同的界限适用于四边形上边缘截面的光谱差距，这使得作者的最新结果和Stauffer成为了通过Tutte的Bievistion的边缘旋转相关的链条的最新结果。
We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations of the $n$-gon and quadrangulations of the sphere, among others). We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge rotation chain is bounded below by an appropriate constant times $n^{-11/2}$. In doing so, we provide a partially new proof of the fact that the same bound applies to the spectral gap of edge flips on quadrangulations, which makes it possible to generalise a recent result of the author and Stauffer to a chain that relates to edge rotations via Tutte's bijection.
