Direct Sampling of Confined Polygons in Linear Time

TL;DR

The paper addresses sampling random equilateral polygons under tight rooted spherical confinement, introducing the CPOP algorithm that leverages symplectic reduction to a moment polytope and achieves linear-time sampling in the edge count $n$ for $R=1$. By mapping the confined polytope to the zig-zag order polytope and exploiting its connection to alternating permutations, the authors obtain explicit formulas for expected chord lengths and reveal precise asymptotics through Entringer-number-based combinatorics. They develop a comprehensive combinatorial framework around augmented zigzag posets $Z_{n,i}$ to link polygon geometry to linear extensions and Entringer numbers, and validate their results with large-scale numerical experiments on turning angles and total curvature, culminating in a conjectured asymptotic form for total curvature in confinement. The work provides both a fast, exact sampling method and deep combinatorial insights that inform the geometry of tightly confined polymers and potential knotting phenomena, with open questions on extending to other radii and further analytic descriptions of key constants.

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.

Figures (10)

• Figure 1: Illustrating the reconstruction map $\alpha: \mathcal{P}_n \times T^{n-3}$ which takes diagonal lengths and dihedral angles to an equilateral polygon. Top left shows the triangulation of an abstract hexagon. Given $d_1, d_2, d_3$ which obey the triangle inequalities \ref{['eq:fan polytope']}, build the four triangles in the triangulation from their side lengths (top right). Given dihedral angles $\theta_1, \theta_2, \theta_3$, we can build a piecewise-linear surface out of these triangles (bottom left). The boundary of this surface is the resulting polygon in space (bottom right).
• Figure 2: Left: Average rejection probabilities when generating 1,000,000 random points in $\mathcal{P}_n(1)$ with $n=4,\dots , 50$ (dots), compared to the asymptotic limit $1-\frac{8}{\pi^2} \approx 0.1894$ (black line). Right: Average number of repeats of the main loop in \ref{['alg:polytope sampling']} when generating 1,000,000 random points in $\mathcal{P}_n(1)$ for the same range of $n$ (dots), compared to the estimate $\frac{\pi^2}{8} \approx 1.2337$ (black line).
• Figure 3: Time per million samples for $n$-gons from $n=500$ to $n=20,\!000$ in steps of $500$. The fitted line has slope $0.102$ (with $R^2 > 0.9999$).
• Figure 4: For (left) $n=20$ and $N=1$ million and (right) $n=20,\!000$ and $N=10,\!000$, we generated $N$ samples of $(d_1,\dots, d_{n-3})$ with \ref{['alg:polytope sampling']}. The plots show the histograms of the resulting $N(n-3)$ numbers (i.e., all the $d_i$ for all the samples, treated as a single pool of numbers), in both cases plotted against the asymptotic density $f(t) = 1-\cos(\pi t)$.
• Figure 5: Hasse diagram for the poset $Z_{n,i}$.

Theorems & Definitions (40)

• Theorem 1: Cantarella--Shonkwiler cantarellaSymplecticGeometryClosed2016
• Theorem 2: Cantarella--Shonkwiler cantarellaSymplecticGeometryClosed2016
• Definition 3
• Theorem 4: Marchal marchalGeneratingRandomAlternating2012
• Theorem 5
• Corollary 6
• Definition 7
• Proposition 8: see Graham, Knuth, and Patashnik grahamConcreteMathematicsFoundation1994 and Stanley stanleySurveyAlternatingPermutations2010
• Definition 9
• Example