连续划分中短路径的直接采样
Direct sampling of short paths for contiguous partitioning
AI总结：
本文提出基于动态规划的算法，可在多项式时间内采样整数格上两点间长度为 n+O(n^{1-δ}) 的近似最短自避行走，并证明该算法能高效均匀采样阿兹特克钻石的连续区域划分，而相应 Glauber 动力学混合时间是指数的。
摘要：
本文提出了一族基于动态规划的算法，用于在整数格 $\mathbb{Z}^2$ 中两点之间采样近似最短的自避行走。我们证明，如果两点之间最短路径的长度为 $n$，那么我们可以在多项式时间内采样长度为 $n+O(n^{1-δ})$ 的路径（自避行走）。作为应用示例，我们将证明，对于将 $\mathbb{Z}^2$ 中阿兹特克钻石划分为两个具有近似紧致周长约束的连续区域，其 Glauber 动力学马尔可夫链具有指数混合时间，而本文提供的算法可用于高效地均匀（且精确地）采样此类划分。
英文摘要：
In this paper, we provide a family of dynamic programming based algorithms to sample nearly-shortest self avoiding walks between two points of the integer lattice $\mathbb{Z}^2$. We show that if the shortest path of between two points has length $n$, then we can sample paths (self-avoiding-walks) of length $n+O(n^{1-δ})$ in polynomial time. As an example of an application, we will show that the Glauber dynamics Markov chain for partitions of the Aztec Diamonds in $\mathbb{Z}^2$ into two contiguous regions with nearly tight perimeter constraints has exponential mixing time, while the algorithm provided in this paper can be used be used to uniformly (and exactly) sample such partitions efficiently.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
