Random convex chains through the lens of analytic combinatorics

TL;DR

The paper studies the random convex chain $T_n$ formed in the triangle $T$ by $n$ i.i.d. points and analyzes the vertex count $f_0(T_n)$. It derives a closed-form bivariate generating function $F(u,z)$ for the distribution of $f_0(T_n)$ by solving a second-order ODE and expressing the solution in terms of the Gaussian hypergeometric function, with $oldsymbol{\alpha}(z)=rac{oldsymbol{ ext{sqrt}{8z+1}}-1}{2}$. Using singularity analysis, it obtains precise cumulant asymptotics, a quantitative central limit theorem with rate, a moderate and large deviation principle for $f_0(T_n)/oldsymbol{ ext{log}n}$, and detailed asymptotics for the probabilities $p_k^{(n)}= ext{P}(f_0(T_n)=k+2)$ across several regimes. The results establish a bridge between stochastic geometry and analytic combinatorics, showing that distributional questions about random polygons can be addressed through analytic properties of generating functions, and extend analytic combinatorics to a genuinely continuous geometric setting.

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 $f_0(T_n)$, the number of vertices of this chain. Our first result gives an explicit expression for the bivariate generating function of the probabilities $\mathbb{P}(f_0(T_n)=k+2)$ 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 $f_0(T_n)$, including a quantitative central limit theorem, a large deviation principle as well as a precise asymptotics for the probabilities $\mathbb{P}(f_0(T_n)=k+2)$. Conceptually, our results establish a novel bridge between stochastic geometry and methods from analytic combinatorics.

Figures (5)

• Figure 1: Decomposition of a random polygon in a square into four convex chains.
• Figure 2: Simulations of a random convex chain generated by $n=10$ (left), $n=50$ (middle) and $n=100$ (right) points.
• Figure 3: Histogram of the probability $p_k^{(n)}=\mathbb{P}(f_0(T_n)=k+2)$ generated by $1000$ runs with $n=5,000$ (left) and $n=50,000$ (right). The red curve is the plot of the Gaussian density function $\varphi_n(x) = \frac{1}{\sigma_n\sqrt{2\pi}} \exp\left(-\frac{(x-\mu_n)^2}{2\sigma_n^2}\right)$, where $\mu_n = \frac{2}{3}\log n \sim \mathbb{E} f_0(T_n)$ and $\sigma_n = \sqrt{\frac{10}{27} \log n} \sim \sqrt{\operatorname{var} f_0(T_n)}$, see Section 3.
• Figure 4: The rate function $I(x)$ for $x\geq 0$ of the large deviation principle in Theorem \ref{['thm:LDP']}. It is strictly convex with a unique minimum of $0$ at $x=2/3$.
• Figure 5: Graph of $|I_n(s+it)|$ for $n=10^6$ and $c=2/3$ (left), $c=3/4$ (middle), and $c=0.8$ (right). The green curve is the section $s\equiv \tilde{\alpha}_*$ and the red curve is the section $s\equiv \alpha_*$.

Theorems & Definitions (40)

• Theorem 2.1: ODE for the bivariate generating function
• proof
• Corollary 2.2: ODE for the bivariate generating function
• proof
• Theorem 2.3: Solution of the ODE for the bivariate generating function
• proof
• Remark 2.4: Alternative representations of the solution
• Theorem 2.5: Asymptotics for the bivariate generating function
• proof
• Corollary 2.6: Asymptotics for the ordinary generating function