2023/10/29 by Jason Cantarella, Cantarella, Jason, Henrik Schumacher +1
Mathematics · Physics and Astronomy · #60D05 #65D18 #82D60 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math.DG #math.PR #msc:60D05 #msc:65D18 #msc:82D60
paper · pdf · doi:10.48550/arxiv.2310.19134
published as SIAM J. Appl. Algebra Geom. 8 (2024) 756-781 · 27 pages, 8 figures
arxiv created 2023/10/29 · arxiv updated 2026/08/03
We present the first algorithm for sampling random configurations of closed n-gons with any fixed edgelengths r1, …, rn in any dimension d which is proved to sample correctly from standard probability measures on these spaces. We generate open n-gons as weighted sets of edge vectors on the unit sphere and close them by taking a Möbius transformation of the sphere which moves the center of mass of the edges to the origin. Using previous results of the authors, such a Möbius transformation can be found in O(n) time. The resulting closed polygons are distributed according to a pushforward measure. The main contribution of the present paper is the explicit calculation of reweighting factors which transform this pushforward measure to any one of a family of standard measures on closed polygon space, including the symplectic volume for polygons in ℝ3. For fixed dimension, these reweighting factors may be computed in O(n) time. Experimental results show that our algorithm is efficient and accurate in practice, and an open-source reference implementation is provided.