Sylvester's question and the Random Acceleration Process

TL;DR

This paper links Sylvester's question about convex polygons formed by random points to the random acceleration process, deriving asymptotic probabilities and revealing new nonanalytic behavior in the large-n limit.

Contribution

It establishes a novel connection between convex polygon probability and the random acceleration process, deriving new asymptotic expansions and nonanalytic terms.

Findings

Asymptotic expansion of log p_n matches known results for first two terms.

Identifies a new nonanalytic term with exponent 1/5 in the expansion.

Shows the polygon is contained in an annulus of width ~ n^{-4/5}.

Abstract

Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = -2n log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms agree with a rigorous result due to Barany. The nonanalyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width \sim n^{-4/5} along the edge of the disk. The distance delta_n of closest approach to the edge is exponentially distributed with average 1/(2n).