Random convex chains through the lens of analytic combinatorics

2025/10/19 by Florian Besau, Besau, Florian, Christoph Thäle +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG) #Pirmary: 52A22 60D05 #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Secondary: 05A15 32A05 33C05 60F10

paper · pdf · doi:10.48550/arxiv.2510.16793

openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

Consider the triangle T with vertices (0,0), (0,1), and (1,0). The lower boundary of the convex hull of (0,1), (1,0), together with n independent uniformly distributed random points in T, is called a random convex chain and denoted by T_n. We study the random variable f₀(T_n), the number of vertices of this chain. Our first result gives an explicit expression for the bivariate generating function of the probabilities ℙ(f₀(T_n)=k+2) in terms of the Gaussian hypergeometric function. Building on this analytic representation, we apply a careful singularity analysis to derive a variety of limit theorems for f₀(T_n), including a quantitative central limit theorem, a large deviation principle as well as a precise asymptotics for the probabilities ℙ(f₀(T_n)=k+2). Conceptually, our results establish a novel bridge between stochastic geometry and methods from analytic combinatorics.

Citations

• On the moments of the volume for random convex chains
• Extremal affine surface areas in a functional setting
• A central limit theorem for random disc-polygons in smooth convex discs
• Variance expansion and Berry-Esseen bound for the number of vertices of a random polygon in a polygon
• Variance Asymptotics and Scaling Limits for Random Polytopes
• Variance asymptotics for random polytopes in smooth convex bodies

Cited by

• On the moments of the volume for random convex chains

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