A Fast Direct Sampling Algorithm for Equilateral Closed Polygons

TL;DR

This paper introduces a fast, numerically stable direct sampling algorithm for equilateral closed polygons, facilitating statistical analysis of ring polymers by efficiently generating representative samples.

Contribution

The authors develop a novel direct sampling method with a new volume formula for equilateral polygon space, improving speed and stability over previous approaches.

Findings

The algorithm is faster than existing methods.

It is numerically stable and reliable.

The volume formula enables efficient sampling.

Abstract

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.