Random Convex Chain Models

Updated 20 October 2025

Random convex chain is a stochastic geometric structure where the convex hull of random points reveals key statistical and combinatorial properties.
It employs models such as random point sets, random walks, and order types to study vertex counts, area distributions, and asymptotic behaviors.
Applications span statistical physics, shape approximation, and randomized interior-point optimization methods for high-dimensional convex bodies.

A random convex chain refers to a sequence or collection of points, trajectories, or structures, typically in the plane or in higher-dimensional space, whose convex hull or geometric configuration is governed by stochastic mechanisms. The concept encompasses discrete models (such as random point sets in convex domains), random walks and stochastic processes, combinatorial chains in order types, and the paper of related geometric statistics such as number of vertices, volume, perimeter, or record statistics.

1. Stochastic Models of Random Convex Chains

Several canonical models have been extensively studied:

Random Point Sets in a Convex Domain: Given a convex domain $T$ (e.g., a triangle or unit square), let %%%%1%%%% be the convex hull of $n$ independent random points (possibly with fixed endpoints). The statistics of $T_n$ (vertex number, area/volume, moments, shape) form the basis for rigorous probabilistic analysis (Gusakova et al., 17 Oct 2025, Gusakova et al., 2020).
Random Walks and Stochastic Processes: In models such as the planar random walk, Pearson walk, Polya lattice walk, or more generally the random acceleration process, the evolving positions create a convex chain whose extremal properties are analyzed using tools from extreme value theory, Cauchy’s formulae, and Markov chain Monte Carlo (Reymbaut et al., 2011, Schawe et al., 2018, Hartmann et al., 2019, Godrèche et al., 26 Apr 2024).
Convex Chains in Order Types: Combinatorial structures such as planar order types and oriented matroids are analyzed for the frequency of long convex chains, with results showing that nearly every large random order type contains a convex $k$ -chain, generalizing Erdős–Szekeres-type theorems (Goaoc et al., 2020).
Higher-Order Convexity: The notion of $k$ -monotone chains, where every $(k+1)$ -tuple of consecutive points is positive in the sense of divided differences, generalizes convex chains ( $k=2$ ) and connects to longest increasing subsequence problems and Hammersley’s process (Ambrus, 2020).

2. Probabilistic and Combinatorial Properties

Vertex Number and Volume Distribution

For random convex polygons in the plane, the number of vertices, $N_n$ , and the normalized volume/area, $V_n$ , satisfy rich probabilistic and combinatorial laws (Gusakova et al., 17 Oct 2025, Gusakova et al., 2020). Key results include:

Probability Generating Function for Vertex Number: $G_n(z)$ satisfies a three-term recurrence:

$G_n(z) = (a_n z + b_n) G_{n-1}(z) - c_n G_{n-2}(z)$

with specified coefficients. Its roots are real and negative, implying $N_n$ is distributed as a sum of independent Bernoulli random variables.

Moments of Volume: Exact formulas for $E[V_n^k]$ via combinatorial recursions and alternative representations involving symmetric polynomials.
Limit Theorems: Central limit theorems, local limit theorems, Berry–Esseen bounds, and mod-Gaussian convergence for $N_n$ , leveraging the PF (Pólya frequency) property of the probability sequence.

Asymptotics

For large $n$ , mean values and variances follow specific scaling laws depending on geometry and underlying distributions:

Uniform points in triangle: asymptotic area $E[V_n] = 1 - (2\log n)/(3n) + O(1/n)$ (Gusakova et al., 17 Oct 2025).
Vertices: $E[N_n] \sim \mathrm{const} \cdot \log n$ (square), $E[N_n] \sim n^{1/3}$ (disk), $E[N_n] \sim \sqrt{\log n}$ (Gaussian) (Godrèche et al., 26 Apr 2024).

3. Random Convex Records

A convex record is a point in a sequence that lies outside the convex hull of all its predecessors. These records form a geometric analogue of univariate record statistics and satisfy asymptotic identities linking their mean number $R_n$ to the expected number of vertices $N_n$ (Godrèche et al., 26 Apr 2024):

$N_n = n(R_n - R_{n-1})$

with explicit scaling laws for various distributions and universal limit theorems for random walks.

4. Geometric and Analytical Methods

Analytical techniques underlying convex chain studies include:

Cauchy’s Formulae: The support function $M(\theta)$ of a convex hull links perimeter and area to extreme value statistics.
Extreme Value Theory: Mapping multivariate hull properties to the distribution and moments of maxima of stochastic processes (Reymbaut et al., 2011, Hartmann et al., 2019).
Markov Chain Monte Carlo: Used for sampling rare events (large deviations in area/perimeter) and reconstructing full distributions (Schawe et al., 2018, Hartmann et al., 2019).

5. Advanced Chaining and Convexity in Function Spaces

For random processes and Gaussian fields indexed by convex sets, chaining bounds (e.g., Dudley’s entropy integral) can be systematically improved when leveraging convexity. Real interpolation methods yield sharper, explicit entropy bounds on suprema, replacing classical entropy numbers $e_n(B)$ with those for “thinner” convex subsets $B_t$ (Handel, 2015). For example, for $\ell_q$ -ellipsoids and octahedra, the optimal chaining bound for the convex hull is constructed via interpolated multiscale nets.

6. Optimization and Interior-Point Methods

A critical application is the design of randomized interior-point algorithms for sampling and optimizing over convex bodies equipped with self-concordant barrier functions (0911.3950):

Dikin Walk: A Markov chain that samples from the barrier-induced geometry. Formally, the local Riemannian norm is given by

$\|v\|_x^2 = D^2F(x)[v, v]$

with proposals drawn from $G_x(y) \propto \exp\left(-\frac{n}{r^2}\|x-y\|_x^2 + V(x)\right)$ and $V(x) = \frac{1}{2}\ln\det D^2F(x)$ .

Affine Invariance: Mixing time and conductance bounds are invariant under affine transformations, eliminating the need for isotropic position.
Mixing Times: For dimension $n$ and barrier complexity $\nu$ , the mixing time from a central point is $O(n\nu \log(1/\varepsilon))$ ; for product bodies, sharper bounds are obtained via isoperimetric inequalities on Riemannian manifolds and product probability measures.
Las Vegas Optimization: The walk is adapted for linear optimization via a projective mapping. It finds $\varepsilon$ -optimal solutions in polynomial time with high probability.

7. Applications and Extensions

Random convex chains and related structures have broad applications:

Statistical Physics: Modeling semi-flexible polymers, persistent random walks, and fluctuations in high-dimensional systems (Reymbaut et al., 2011, Hartmann et al., 2019).
Stochastic Geometry and Shape Approximation: Exact formulas and asymptotics support algorithms for shape recognition, boundary estimation, and convex hull computation (cf. optimal convex layer algorithms (Rufai et al., 2017)).
Combinatorics and Probability: Limit laws, recurrence relations, and PF property-inspired central limit behaviors inform both theoretical and applied studies.
Optimization and Interior-Point Methods: Barrier-induced chains yield efficient randomized algorithms with strong theoretical guarantees for sampling, volume computation, and optimization (0911.3950).

8. Open Problems and Future Research Directions

Current lines of research include:

Determining exact constants for longest $k$ -monotone chains and establishing limit shape results for higher-order monotonicity (Ambrus, 2020).
Extending probabilistic and combinatorial methods to other convex bodies and higher-dimensional polytopes.
Bridging analytic approaches (orthogonal polynomials, mod-Gaussian convergence) with combinatorial models (order types, matroids).
Developing sharper mixing and conductance bounds for random walks and Markov chains on convex bodies of high complexity and in non-Euclidean settings. A plausible implication is that further exploration of product Riemannian manifolds and isoperimetric constants will improve both sampling and optimization algorithms for large-scale convex sets.

This synthesis delineates the central models, probabilistic results, combinatorial identities, and analytical methods in the paper of random convex chains, referencing specific papers for their contributions to discrete, geometric, and algorithmic aspects.