Random geometry and convexity Study of random polytopes Pierre - - PowerPoint PPT Presentation

Random geometry and convexity

Study of random polytopes

Pierre Calka

19 October 2016, IHP 2nd French-Russian conference Random Geometry and Physics

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Point process

Point process : random locally finite set of Rd

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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Point process without interaction

• B1

B2 B3 B4

Binomial point process µ probability measure on Rd X1, · · · , Xn i.i.d. with law µ E := {X1, · · · , Xn} ◮ #(E ∩ B1) binomial r.v. (n, µ(B1)) ◮ #(E ∩ B1), · · · , #(E ∩ Bℓ) not independent Poisson point process µ σ-finite measure on Rd µ := intensity of the process P locally finite set s.t. ◮ #(P ∩ B1) Poisson r.v. (µ(B1)) ◮ #(P ∩ B1), · · · , #(P ∩ Bℓ) independent

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Uniform case

Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K K n := Conv(X1, · · · , Xn), n ≥ 1 K 50, K disk K 50, K square

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Uniform case

Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K K n := Conv(X1, · · · , Xn), n ≥ 1 K 100, K disk K 100, K square

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Uniform case

Binomial model K := convex body of Rd (Xk,k ∈ N∗):= independent and uniformly distributed in K K n := Conv(X1, · · · , Xn), n ≥ 1 K 500, K disk K 500, K square

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Uniform case

Poisson model K := convex body of Rd Pλ, λ > 0:= Poisson point process of intensity measure λdx Kλ := Conv(Pλ ∩ K) K 500, K disk K 500, K square

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Gaussian case

Poisson model ϕd(x) :=

1 (2π)d/2 e−x2/2, x ∈ Rd, d ≥ 2

Pλ, λ > 0:= Poisson point process of intensity measure λϕd(x)dx Kλ := Conv(Pλ) K100 K500

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Plan

Random polytopes: survey and new variance calculations Cases of the disk and the square: sketch of proof by scaling limit Poisson-Voronoi tessellation and isolated points Based on joint works with Joseph Yukich (Lehigh University, USA) and also Tomasz Schreiber (Toru´ n University, Poland), Yann Demichel & Nathana¨ el Enriquez (Paris Ouest)

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Plan

Random polytopes: survey and new variance calculations Known expectation asymptotics Known second-order asymptotic results Main new results: exact limiting variances Cases of the disk and the square: sketch of proof by scaling limit Poisson-Voronoi tessellation and isolated points

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Known expectation asymptotics

fk(·) := number of k-dimensional faces, Vol(·) := volume Efron’s relation (1965)

Ef0(K n) = n

• 1 − EVol(K n−1)

Vol(K)

• Uniform, K smooth

E[fk(Kλ)] ∼

λ→∞ cd,k

• ∂K κ

1 d+1

s

ds λ

d−1 d+1

κs := Gaussian curvature of ∂K

Uniform, K polytope E[fk(Kλ)] ∼

λ→∞ c′

d,kF(K) logd−1(λ)

F(K) := number of flags of K

Gaussian E[fk(Kλ)] ∼

λ→∞ c′′

d,k log d−1 2 (λ)

• A. R´

enyi & R. Sulanke (1963), H. Raynaud (1970), R. Schneider & J. Wieacker (1978), F. Affentranger & R. Schneider (1992)

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Known second-order asymptotic results

◮ Central limit theorems ◮ Variance bounds Uniform, K smooth Var[fk(Kλ)] =

λ→∞ Θ(λ

d−1 d+1 )

Uniform, K polytope Var[fk(Kλ)] =

λ→∞ Θ(logd−1(λ))

Gaussian Var[fk(Kλ)] =

λ→∞ Θ(log

d−1 2 (λ))

• M. Reitzner (2005), I. B´

ar´ any & V. Vu (2007), I. B´ ar´ any & M. Reitzner (2010)

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Main new results: exact limiting variances

Uniform, K smooth Var[fk(Kλ)] ∼

λ→∞ cd,k

• ∂K κ

1 d+1

s

ds λ

d−1 d+1

κs := Gaussian curvature of ∂K

Uniform, K simple polytope Var[fk(Kλ)] ∼

λ→∞ c′

d,kf0(K) logd−1(λ)

Gaussian Var[fk(Kλ)] ∼

λ→∞ c′′

d,k log d−1 2 (λ)

Remarks ◮ Depoissonnization in the uniform smooth and Gaussian cases ◮ Central limit theorems ◮ Uniform in the ball and Gaussian cases: invariance principle for the volume

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Plan

Random polytopes: survey and new variance calculations Cases of the disk and the square: sketch of proof by scaling limit Calculation of the variance of fk(Kλ) Rescaling Floating body Dual characterization of extreme points Action of the scaling transform Convergence of the covariances of scores Poisson-Voronoi tessellation and isolated points

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Calculation of the expectation fk(Kλ)

Decomposition E[fk(Kλ)] = E  

x∈Pλ

ξ(x, Pλ)   ξ(x, Pλ) :=

• 1

k+1#k-face containing x

if x extreme if not Mecke-Slivnyak’s formula E[fk(Kλ)] = λ

• D

E[ξ(x, Pλ ∪ {x})]dx

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Calculation of the variance of fk(Kλ)

Var[fk(Kλ)] = E  

x∈Pλ

ξ2(x, Pλ) +

• x=y∈Pλ

ξ(x, Pλ)ξ(y, Pλ)   − (E[fk(Kλ)])2 = λ

• D

E[ξ2(x, Pλ ∪ {x})]dx + λ2

• D2 E[ξ(x, Pλ ∪ {x, y})ξ(y, Pλ ∪ {x, y})]dxdy

− λ2

• D2 E[ξ(x, Pλ ∪ {x})]E[ξ(y, Pλ ∪ {y})]dxdy

= λ

• D

E[ξ2(x, Pλ ∪ {x})]dx + λ2

• D2 Cov(ξ(x, Pλ ∪ {x}), ξ(y, Pλ ∪ {y}))dxdy

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Rescaling

Question Limits of E[ξ(x, Pλ)] and Cov(ξ(x, Pλ), ξ(y, Pλ))? Definition of limit scores in a new rescaled space Construction of the scaling transform Requires to understand the typical shape of Kλ: the floating body.

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Floating body

V (x) := inf{Vol(K ∩ H+) : H+ half-plane containing x}, x ∈ K Floating body: K(V ≥ t) := {x ∈ K : V (x) ≥ t} K(V ≥ t) is a convex body and K(V ≥ 1/λ) is close to Kλ.

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Floating body

V (x) := inf{Vol(K ∩ H+) : H+ half-plane containing x}, x ∈ K Floating body : K(V ≥ t) := {x ∈ K : V (x) ≥ t} K(V ≥ t) is a convex body and K(V ≥ 1/λ) is close to Kλ. D(V ≥ 1/λ) = (1 − f (λ))D

f (λ) ∼ cλ− 2

3

K(V ≥ 1/λ) = {(x1, x2) : x1x2 ≥

1 2λ}

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Comparison between Kλ and the floating body

◮ Expectation

B´ ar´ any & Larman (1988):

cVol(K(V ≤ 1/λ)) ≤ Vol(K)−E[Vol(Kλ)] ≤ CVol(K(V ≤ 1/λ)) ◮ Variance

B´ ar´ any & Reitzner (2010):

cλ−1Vol(K(V ≤ 1/λ)) ≤ Var[Vol(Kλ)] ◮ Sandwiching

B´ ar´ any & Reitzner (2010b):

P[∂Kλ ⊂ [K(V ≥ s) \ K(V ≥ T)]] = O

• (log(λ))−16

s :=

c λ(log(λ))17 , T := c′ log(log(λ)) λ

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Construction of the scaling transform

Rules of the scaling transform ◮ One depth-coordinate provided by the family of floating bodies: λ

2 3(1 − r) for the disk, λx1x2 for the quadrant

◮ One spatial-coordinate on the corresponding floating body ◮ A rescaling to guarantee Θ(1) points per unit area K = D T λ :

• D \ {0}

− → R × R+ (r, θ) − → (λ

1 3θ, λ 2 3(1 − r))

K = (0, ∞)2 T (λ) : (0, ∞)2 − → L × R (x1, x2) − →

• projL(log(x)), 1

2 log(λx1x2)

• where L = {(x1, x2) ∈ R2 : x1 + x2 = 0}

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Dual characterization of extreme points

Each point x generates a petal S(x), i.e. the set of all tangency points of curves K(V = t

λ), t > 0, with the lines containing x.

x is extreme if and only if its petal is not fully covered by the other petals.

Bd

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Action of the scaling transform in the disk

Π↑ := {(v, h) ∈ R × R+ : h ≥ v 2

2 }, Π↓ := {(v, h) ∈ R × R+ : h ≤ − v 2 2 }

Half-plane Translate of Π↓ Boundary of the hull Union of portions of down-parabolas Petal Translate of ∂Π↑ Extreme point (x + Π↑) not covered Process Pλ Process P of intensity dvdh Score ξ(x, Pλ) Score ξ(∞)((v, h), P)

Bd

− →

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Action of the scaling transform in the quadrant

G(v) := log

• ch( v

√ 2)

• ,

v ∈ V Π↑ := {(v, h) ∈ R × R : h ≥ G(v)}, Π↓ := {(v, h) ∈ R × R : h ≤ −G(v)}

Floating bodies Horizontal half-planes Boundary of the hull Union of portions of pseudo-cones Petal Translate of ∂Π↑ Extreme point (x + Π↑) not covered Process Pλ Process P of intensity √ 2e2hdvdh Score ξ(x, Pλ) Score ξ(∞)((v, h), P) − →

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Convergence of the covariances of scores

◮ Scores ξ(λ)(·, ·) defined by ξ(λ)(T λ(x), T λ(Pλ)) := ξ(x, Pλ) ◮ Scores ξ(∞)(w, P), w = (v, h) ∈ R × R, defined in the limit model ◮ These scores stabilize exponentially. Stabilization radius R(w, P) : smallest r > 0 such that ξ(∞)(w, P) = ξ(∞)(w, P ∩ Cyl(w, r)) P[R(w, P) > t] ≤ ce− t

c ,

t > 0 ◮ Convergence point by point in the integral E[ξ(λ)(w, T λ(Pλ))] → E[ξ(∞)(w, P)] and Cov(ξ(λ)(w, T λ(Pλ)), ξ(λ)(w′, T λ(Pλ))) → Cov(ξ(∞)(w, P), ξ(∞)(w′, P))

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Outline

Random polytopes: survey and new variance calculations Cases of the disk and the square: sketch of proof by scaling limit Poisson-Voronoi tessellation and isolated points Poisson-Voronoi tessellation Approximation of a convex body from the outside

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Poisson-Voronoi tessellation

◮ Pλ Poisson point process in Rd of intensity λ ◮ For every nucleus x ∈ Pλ, the cell associated is C(x|Pλ) := {y ∈ Rd : y − x ≤ y − x′ ∀x′ ∈ Pλ} ◮ Tessellation: set of cells C(x|Pλ)

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Probability to belong to the zero-cell

C0 K F0(K)

K convex body containing 0, C0 Voronoi cell C(0|Pλ ∪ {0}) P(K ⊂ C0) = exp(−λVol(F0(K))) where Vd is the volume and F0(K) = ∪x∈KB(x, x) flower of K

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Approximation of a convex body from the outside

◮ K convex body in the plane R2 ◮ An origin 0 is chosen inside K (not intrinsic!) ◮ Pλ conditional on the event that the Voronoi tessellation associated with Pλ ∪ {0} does not intersect K ◮ Question. Limit of the geometric characteristics of the cell Kλ ⊃ K when λ → ∞?

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Approximation of a convex body from the outside

• ◮ K convex body in the plane R2

◮ An origin 0 is chosen inside K (not intrinsic!) ◮ Pλ conditional on the event that the Voronoi tessellation associated with Pλ ∪ {0} does not intersect K ◮ Question. Limit of the geometric characteristics of the cell Kλ ⊃ K when λ → ∞?

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Approximation of a convex body from the outside

• ◮ K convex body in the plane R2

◮ An origin 0 is chosen inside K (not intrinsic!) ◮ Pλ conditional on the event that the Voronoi tessellation associated with Pλ ∪ {0} does not intersect K ◮ Question. Limit of the geometric characteristics of the cell Kλ ⊃ K when λ → ∞?

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Approximation of a convex body from the outside

• ◮ K convex body in the plane R2

◮ An origin 0 is chosen inside K (not intrinsic!) ◮ Pλ conditional on the event that the Voronoi tessellation associated with Pλ ∪ {0} does not intersect K ◮ Question. Limit of the geometric characteristics of the cell Kλ ⊃ K when λ → ∞?

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Approximation of a convex body from the outside

◮ K convex body in the plane R2 ◮ An origin 0 is chosen inside K (not intrinsic!) ◮ Pλ conditional on the event that the Voronoi tessellation associated with Pλ ∪ {0} does not intersect K ◮ Question. Limit of the geometric characteristics of the cell Kλ ⊃ K when λ → ∞?

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Approximation of a convex body from the outside

• K smooth

E[f0(Kλ)] ∼

λ→∞ 223− 4

3 Γ

• 2

3 ∂K

κ

2 3

s s, ns

1 3 ds λ

1 3

κs := curvature at s ns := normal vector at s

K polygon E[f0(Kλ)] ∼

λ→∞ 2 · 3−1f0(K) log(λ)

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Reverse question

R large planar region containing 0, Pλ point process outside R

• Question. Geometry of the cell containing 0 when λ → ∞?

◮ Consider the maximal flower F included in R and the unique convex body K included in R and whose flower is F ◮ Apply the previous results with K and 0

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