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.

翻译：本文提出一种在三维空间中采样紧密受限随机等边闭合多边形的算法，其运行时间与边数呈线性关系。利用辛几何方法，此类多边形的采样可简化为对矩多面体的采样；在我们的受限模型中，该多面体从组合学视角展现出高度自然的特性。这种与组合学的关联不仅催生了快速采样算法，还推导出顶点到原点期望距离的显式表达式。我们运用该算法探究受限多边形的期望总曲率，进而提出关于总曲率渐近行为的精确猜想。