Direct Sampling of Confined Polygons in Linear Time

TL;DR

This paper introduces a linear-time algorithm for sampling confined equilateral polygons in three-dimensional space, leveraging symplectic geometry and combinatorics to analyze their properties.

Contribution

The authors develop a novel, efficient sampling method for confined polygons using geometric and combinatorial insights, providing explicit formulas and conjectures.

Findings

Sampling time is linear in the number of edges.

Explicit formulas for expected vertex distances to the origin.

Conjecture on asymptotics of total curvature.

Abstract

We present an algorithm for sampling tightly confined random equilateral closed polygons in three-space which has runtime linear in the number of edges. Using symplectic geometry, sampling such polygons reduces to sampling a moment polytope, and in our confinement model this polytope turns out to be very natural from a combinatorial point of view. This connection to combinatorics yields both our fast sampling algorithm and explicit formulas for the expected distances of vertices to the origin. We use our algorithm to investigate the expected total curvature of confined polygons, leading to a very precise conjecture for the asymptotics of total curvature.