Random convex chains through the lens of analytic combinatorics

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.

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