Direct sampling of short paths for contiguous partitioning

TL;DR

This paper introduces dynamic programming algorithms for efficiently sampling nearly-shortest self-avoiding walks on a lattice, enabling uniform sampling of complex partitions with tight constraints, and analyzes their mixing times.

Contribution

It presents a novel polynomial-time method for sampling nearly-shortest self-avoiding walks and applies it to efficiently generate partitions with tight perimeter constraints.

Findings

Sampling paths of length n+O(n^{1- ext{delta}}) in polynomial time.

The Glauber dynamics Markov chain for certain partitions has exponential mixing time.

The proposed algorithm can uniformly and exactly sample such partitions efficiently.

Abstract

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.