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 $n+O(n^{1-\delta})$ 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.

翻译：本文提出了一系列基于动态规划的算法，用于在整数格点 $\mathbb{Z}^2$ 中两点之间采样近似最短的自避行走路径。我们证明，若两点之间最短路径的长度为 $n$，则可以在多项式时间内采样长度为 $n+O(n^{1-\delta})$ 的路径（自避行走）。作为应用实例，我们将证明在 $\mathbb{Z}^2$ 中，针对具有近似紧致周长约束的亚兹特克钻石分区为两个连通区域的Glauber动力学马尔可夫链，其混合时间呈指数增长；而本文提出的算法可用于高效地均匀（且精确地）采样此类分区。