Direct Sampling of Confined Polygons in Linear Time
For researchers in computational geometry and polymer physics, this provides an efficient sampling method for confined polygons, enabling new investigations into geometric properties.
The paper presents a linear-time algorithm for sampling tightly confined random equilateral closed polygons in 3D, using symplectic geometry and combinatorial connections. It also provides explicit formulas for expected vertex distances and investigates total curvature asymptotics, leading to a precise conjecture.
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.