Random convex chains through the lens of analytic combinatorics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914243386802176 |
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| author | Besau, Florian Thäle, Christoph |
| author_facet | Besau, Florian Thäle, Christoph |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random convex chains through the lens of analytic combinatorics Besau, Florian Thäle, Christoph Probability Combinatorics Metric Geometry Pirmary: 52A22 60D05, Secondary: 05A15 32A05 33C05 60F10 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. |
| title | Random convex chains through the lens of analytic combinatorics |
| topic | Probability Combinatorics Metric Geometry Pirmary: 52A22 60D05, Secondary: 05A15 32A05 33C05 60F10 |
| url | https://arxiv.org/abs/2510.16793 |