Threshold phenomena for high-dimensional random polytopes

Bonnet, Gilles; Chasapis, Giorgos; Grote, Julian; Temesvari, Daniel; Turchi, Nicola

doi:10.1142/s0219199718500384

Cited by 26 publications

(40 citation statements)

References 17 publications

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“…that is, the threshold is super-exponential. The latter result has been extended in [5] to the class of so-called beta distributions which have recently attracted a considerable attraction in stochastic geometry [1,6,10,[16][17][18][19][20][21][22][23]25,27]. These distributions are described by their density, which is proportional to (1 − ||x|| 2 ) 1(||x|| < 1), where || ⋅ || denotes the Euclidean norm and where > −1 is a parameter which might depend on d. For example, = 0 generates the uniform distribution on the Euclidean unit ball.…”

Section: Introductionmentioning

confidence: 77%

“…Remark 1. Taking into account both Theorem 3.1 and the threshold established in [5], we can state that, for any sequences of natural numbers n = n(d) and real numbers = ( ) ≥ −1,…”

Section: The Main Results and Related Corollariesmentioning

confidence: 99%

“…In order to prove Theorem 3.1, we need to show that the limit (6) holds for any fixed x ∈ (0, ∞), any sequence ( ) ≥ −1 and any sequence n(d) satisfying condition (5). In a first step we show that it is enough to consider sequences n(d) satisfying a more restrictive condition than (5) Proof.…”

Section: Proofsmentioning

confidence: 99%

“…In the last few years there has been an increasing interest in the study of high-dimensional random convex hulls, see [5,7,9,11,14,15,26] for example. Older works include [3,8,24,25].…”

Section: Introductionmentioning

confidence: 99%

“…We extend this family of distributions by defining the beta distribution with = −1 to be the uniform distribution on the unit sphere. For such distributions, that we denote here by , it was shown in [5] that…”

Section: Introductionmentioning

confidence: 99%

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Phase transition for the volume of high‐dimensional random polytopes

Bonnet

Kabluchko

Turchi

2020

Random Struct Algorithms

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The beta polytope P n, is the convex hull of n i.i.d. random points distributed in the unit ball of R according to a density proportional to (1 − ||x|| 2 ) if > −1 (in particular, = 0 corresponds to the uniform distribution in the ball), or uniformly on the unit sphere if = −1. We show that the expected normalized volumes of high-dimensional beta polytopes exhibit a phase transition and we describe its shape. We derive analogous results for the intrinsic volumes of beta polytopes and, when = 0, their number of vertices.

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Section: Introductionmentioning

confidence: 77%

“…Remark 1. Taking into account both Theorem 3.1 and the threshold established in [5], we can state that, for any sequences of natural numbers n = n(d) and real numbers = ( ) ≥ −1,…”

Section: The Main Results and Related Corollariesmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

“…In the last few years there has been an increasing interest in the study of high-dimensional random convex hulls, see [5,7,9,11,14,15,26] for example. Older works include [3,8,24,25].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Phase transition for the volume of high‐dimensional random polytopes

Bonnet

Kabluchko

Turchi

2020

Random Struct Algorithms

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Facets of spherical random polytopes

Bonnet

O'Reilly

2022

Mathematische Nachrichten

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Facets of the convex hull of 𝑛 independent random vectors chosen uniformly at random from the unit sphere in ℝ 𝑑 are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension increases. Regimes for 𝑛 and 𝑑 with different asymptotic behavior of these quantities are identified and asymptotic formulas in each case are established. Extensions of several known results in fixed dimension to the case where dimension tends to infinity are described.

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Threshold for the expected measure of random polytopes

2023

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Let $$\mu $$ μ be a log-concave probability measure on $${\mathbb R}^n$$ R n and for any $$N>n$$ N > n consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ K N = conv { X 1 , … , X N } , where $$X_1,X_2,\ldots $$ X 1 , X 2 , … are independent random points in $${\mathbb R}^n$$ R n distributed according to $$\mu $$ μ . We study the question if there exists a threshold for the expected measure of $$K_N$$ K N . Our approach is based on the Cramer transform $$\Lambda _{\mu }^{*}$$ Λ μ ∗ of $$\mu $$ μ . We examine the existence of moments of all orders for $$\Lambda _{\mu }^{*}$$ Λ μ ∗ and establish, under some conditions, a sharp threshold for the expectation $${\mathbb {E}}_{\mu ^N}[\mu (K_N)]$$ E μ N [ μ ( K N ) ] of the measure of $$K_N$$ K N : it is close to 0 if $$\ln N\ll {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≪ E μ ( Λ μ ∗ ) and close to 1 if $$\ln N\gg {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≫ E μ ( Λ μ ∗ ) . The main condition is that the parameter $$\beta (\mu )=\textrm{Var}_{\mu }(\Lambda _{\mu }^{*})/({\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*}))^2$$ β ( μ ) = Var μ ( Λ μ ∗ ) / ( E μ ( Λ μ ∗ ) ) 2 should be small.

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Threshold phenomena for high-dimensional random polytopes

Cited by 26 publications

References 17 publications

Phase transition for the volume of high‐dimensional random polytopes

Phase transition for the volume of high‐dimensional random polytopes

Facets of spherical random polytopes

Threshold for the expected measure of random polytopes

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