Poisson-Voronoi graph on a Riemannian manifold

Poisson-Voronoi graph on a Riemannian manifold Pierre Calka October 3, 2018 Random graphs and its applications for networks Saint-Étienne

Poisson-Voronoi tessellation in R n P λ homogeneous Poisson point process in R n of intensity λ For every nucleus x P λ, associated cell C(x, P λ ) := {y R n : y x y x x P λ }

Typical Poisson-Voronoi cell Typical cell C chosen uniformly among all cells 1 E(f (C)) := lim r N r E(f (C)) := 1 λvol (Rn) (B) E x P λ B (Rn) (o,r) ( x P λ B f (C(x, P λ )) a.s. f (C(x, P λ )) for every translation-invariant functional f Theorem (Slivnyak) C D = C(o, P λ {o}) ), B B(R n ) 0

Mecke s formula For any l 1 and any non-negative measurable function f, ( ) E f ({x 1,..., x l }, P λ ) {x 1,...,x l } P λ = λl E(f ({x 1,..., x l }, P λ {x 1,..., x l }))dx 1... dx l l! Characterization of the Poisson point process

Two mean calculations in R 2 Mean area E(vol (R2) (C)) = P(x C(o, P λ {o})dx = 2π R 2 Mean number of vertices 0 e λπr 2 rdr = 1 λ N ( ) := number of vertices, B(x, y, z) := circumscribed disk of the triangle xyz E(N (C)) = E {x 1,x 2 } P λ 1 {B(o,x1,x 2 ) P λ = } θ2 x1 = λ2 θ1 P(B(o, x 1, x 2 ) P λ = )dx 1 dx 2 r 2 = λ2 )(B(o,x1 e λvol(r2,x 2 )) ϕ o dx 1 dx 2 2 = 4πλ 2 e λπr 2 r 3 dr sin( θ 1 (0,2π) 2 2 ) sin(θ 2 2 ) sin(θ 2 θ 1 ) dθ 1 dθ 2 2 = 6 x2

Poisson-Voronoi tessellation on a Riemannian manifold Mean asymptotics for the number of vertices of a Voronoi cell Limit theorems for the empirical mean of the number of vertices Joint work with Aurélie Chapron (Paris Nanterre) & Nathanaël Enriquez (Paris-Sud)

Plan Mean asymptotics for the number of vertices of a Voronoi cell Model Context Main result Probabilistic proof of the Gauss-Bonnet theorem Sketch of proof of the main result Limit theorems for the empirical mean of the number of vertices

Model c R. Kunze (1985) M Riemannian manifold of dimension n endowed with the distance d (M) and the volume measure vol (M) P λ Poisson point process in M of intensity measure λvol (M) Voronoi cell associated with x P λ C (M) (x, P λ ) = {y M : d (M) (x, y) d (M) (x, y) x P λ } Aim Study the mean geometrical characteristics of C (M) x o,λ := C (M) (x 0, P λ {x 0 }), x 0 M fixed

Context Influence of the local geometry of M around x 0 on the geometry of C (M) (x 0, P λ {x 0 }) Need to do estimations at high intensity Conversely, information provided by the tessellation on the geometry of M, both local (curvatures at x 0 ) and global (Gauss-Bonnet) Need to show limit theorems in order to do a statistical estimation

Previous works S 2 k 2-sphere of radius 1/k H 2 k hyperbolic plane of curvature k 2 c J. Diaz-Polo & I. Garay (2014) c I. Benjamini, E. Paquette & J. Pfeffer (2014)

Previous works S 2 k 2-sphere of radius 1/k H 2 k hyperbolic plane of curvature k 2 Mean area E(vol (S2 k ) (C (S2 k ) x 0,λ ))) = 1 λ (1 e 4πλ k n ) Mean area E(vol (Hn k ) (C (H2 k ) x 0,λ ))) = 1 λ Mean number of vertices ( ) E N (C (S2 k ) x 0,λ ) ( ) = 6 3k2 πλ + 4πλ e k 2 6 + 3k2 πλ Mean number of vertices Y. Isokawa (2000) ( ) E N (C (H2 k ) x 0,λ ) = 6 + 3k2 πλ. R. E. Miles (1971)

Main result General assumptions on M: sectional curvatures bounded, global injectivity radius... Sc (M) x 0 := scalar curvature of M at x 0 E(N (C (M) x 0,λ )) = e n d n Sc (M) 1 x 0 λ λ 2 n where e n = E(N (C (Rn) (o, P λ {o}))) and d n are explicit. ( ) 1 + o λ 2 n ( ) E(vol (M) (C (M) x 0,λ )) = 1 λ λ + o 1 λ 1+ n 2

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) =? 1 4π M Sc (M) x dvol (M) (x)

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = F E + V by Euler s formula applied to the Poisson-Voronoi graph F:= # faces, E:= # edges, V:= # vertices

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = F E + V 2E = 3V

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = F 1 2 V

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = E(F ) 1 2 E(V )

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = λvol (M) (M) 1 2 E(V )

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = λvol (M) (M) 1 6 E N (C (M) x,λ ) x P λ

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = λvol (M) (M) λ 6 E(N (C (M) x,λ ))dvol(m) (x)

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = λvol (M) (M) λ 6 E(N (C (M) x,λ ))dvol(m) (x) 6 3Sc(M) x 2πλ +o 0 1 @1A λ

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = λvol (M) (M) λvol (M) (M) + 1 4π Sc (M) x vol (M) (x) + o(1)

Probabilistic proof of the Gauss-Bonnet theorem M 2-dimensional compact manifold, χ(m):= Euler characteristic of M χ(m) = 1 4π Sc (M) x dvol (M) (x)

Main result General assumptions on M: sectional curvatures bounded, global injectivity radius... Sc (M) x 0 := scalar curvature of M at x 0 E(N (C (M) x 0,λ )) = e n d n Sc (M) 1 x 0 λ λ 2 n where e n = E(N (C (Rn) (o, P λ {o}))) and d n are explicit. ( ) 1 + o λ 2 n

Sketch of proof: preliminary calculation E(N (C (M) x 0,λ )) = E λn n! = λn n! {x 1,...,x n} P λ B circumball of x 0,...,x n 1 {B Pλ = } P(CircumBall(x 0,..., x n ) P λ = )dvol (M) (x 1 )... dvol (M) (x n ) B (M) (x 0,ε) n e λvol(m) (CircumBall(x0,...,x n)) dvol (M) (x 1 )... dvol (M) (x n ) B (M) (x 0,ε) n Aim Estimates when x 1,..., x n x 0 of vol (M) (CircumBall(x 0,..., x n )) and dvol (M) (x 1 )... dvol (M) (x n ) before applying Laplace s method

Sketch of proof: change of variables x0 u r Tx0 M r M exp x0 (ru) Exponential map For each unit-vector u T x0 M, γ : r exp x0 (ru) is the unique geodesic emanating from x 0 with speed 1 and γ (0) = u. x0 u0 r M x1 r u1 exp x0 (ru) u2 r x2 Change of variables x i = exp expx0 (ru 0 )(ru i ), i = 1,..., n

Sketch of proof: asymptotics for the integrand Expansion of the Jacobian: Blaschke-Petkantschin formula dvol (M) (x 1 )... dvol (M) (x n ) = J drdvol (Sn 1) (u 0 )... dvol (Sn 1) (u n ), ( ) n J = n! (u 0,..., u n ) r n2 1 i=0 Ric(M) x 0 (u i ) r n2 +1 + o(r n2 +1 ) 6 Ric (M) x 0 (u i ):= Ricci curvature at x 0 in direction u i, (u 0,..., u n ):= vol (Rn) (Conv(u 0,..., u n )) Volume of small balls Theorem (Bertrand-Diguet-Puiseux) vol (M) (B (M) (x 0, r)) = vol (Rn) (B (Rn) (o, r)) ( ) 1 Sc(M) x 0 6(n + 2) r 2 + o(r 2 )

Sketch of proof: expansion of the Jacobian Each partial derivative is a Jacobi field J along a geodesic γ which satisfies both the Jacobi equation J = R γ(t) (γ (t), J(t))γ (t) and the Rauch comparison theorem J(t) γ(t) = t K (M) γ(0) (J (0), γ (0)) t 3 + o(t 3 ). 6 R γ(t) (, ) := curvature tensor at γ(t), K (M) γ(0) (, ) := sectional curvature at γ(0) Expansion of each entry of the Jacobian matrix and careful calculation of the determinant

Further results Density of vertices in a fixed direction Same quantities for sectional tessellations Exact formulae in the case of the constant curvature

Plan Mean asymptotics for the number of vertices of a Voronoi cell Limit theorems for the empirical mean of the number of vertices Motivation Mean asymptotics Variance asymptotics Central limit theorem Estimation of Sc (M) x 0

Motivation Aim Use the knowledge of the Poisson-Voronoi graph to recover the curvature at x 0 Replace E(N (C (M) x 0,λ )) with an empirical mean around x 0 N (M) λ := x B(x 0,λ β ) P λ N (C (M) (x, P λ )), 0 < β < 1/n Show expectation, variance asymptotics and a CLT for N (M) λ Deduce the construction of an estimator of Sc (M) x 0 asymptotically unbiased, consistent and normal. which is

Mean asymptotics of N (M) λ E(N (M) λ ) = λ κ n λ 1 nβ (e n d n Sc (M) x 0 λ 2 n ) + o(λ 1 nβ 2 n ) where κ n := volume of the n-dimensional unit-ball Sketch of proof ξ(x, P λ ) := N (C (M) (x, P λ )) E(N (M) λ ) = λ x B (M) (x 0,λ β ) E(ξ(x, P λ))dvol (M) (x) We require the uniformity of the two-term expansion of E(ξ(x, P λ )) with respect to x.

Variance asymptotics for N (M) λ 1 3n < β < 1 n Var(N (M) λ ) λ κ n λ 1 nβ σ where κ n := volume of the n-dimensional unit-ball and σ := E(N (C (Rn ) o,1 )2 ) + Cov(N (C (Rn) (x, P 1 {x})), N (C (Rn ) o,1 ))dx. (N (M) λ E(N (M) λ )) of order Var(N (M) λ ), i.e. λ 1 2 (1 nβ) must be negligible in front of the second term of E(N (M) λ ), i.e. λ 1 nβ 2 n.

Sketch of proof for the variance asymptotics Var(N (M) λ ) = E ξ 2 (x, P λ ) + ξ(x, P λ )ξ(y, P λ ) E(N (M) x x y = λ E(ξ 2 (x, P λ {x}))dvol (M) (x) B (M) (x 0,λ β ) λ )2 + λ 2 B (M) (x 0,λ β ) 2 E(ξ(x, P λ {x, y})ξ(y, P λ {x, y}))dvol (M) (x)dvol (M) (y) λ 2 B (M) (x 0,λ β ) 2 E(ξ(x, P λ {x}))e(ξ(y, P λ {y}))dvol (M) (x)dvol (M) (y) = λ E(ξ 2 (x, P λ {x}))dvol (M) (x) B (M) (x 0,λ β ) + λ 2 B (M) (x 0,λ β ) 2 Cov(ξ(x, P λ {x}), ξ(y, P λ {y}))dvol (M) (x)dvol (M) (y)

Sketch of proof for the variance asymptotics Question Limits of E(ξ 2 (x, P λ )) and Cov(ξ(x, P λ ), ξ(y, P λ ))? Application of the inverse of the exponential map at x 0, then a rescaling by λ 1 n in the tangent space identified with R n Comparison of the obtained tessellation with an Euclidean Poisson-Voronoi tessellation and convergence of the scores Each score has a localization radius R, i.e. ξ (λ) (x, P λ ) = ξ (λ) (x, P λ B (M) (x, R)) where P{R > λ 1 n t} ce tn c uniformly in x and λ.

Central limit theorem for N (M) λ 1 3n < β < 1 n sup t R P ( N (M) λ E(N(M) λ ) Var(N (M) λ ) t ) P(N (0, 1) t) cλ 1 2 (1 nβ). Sketch of proof F functional of P λ with E(F ) = 0 and Var(F ) = 1 Estimate of d K (F, N (0, 1)) involving integrals of moments of consecutive Malliavin derivatives of F where D x F = F (P λ {x}) F (P λ ) G. Last, G. Peccati & M. Schulte (2016) Uniform bounds for the moments of D x N (M) λ and D 2 x,yn (M) λ

Estimation of Sc (M) x 0 0 < β < 1 n 4 n 2 and n 6 Ŝc λ (x 0 ) := λ 2 n d n (e n 1 λvol (M) (B (M) (x 0,λ β )) N(M) λ ) consistent and normal estima- is an asymptotically unbiased, tor of Sc (M) x 0. Ŝc λ (x 0 ) := λ 2 n d n (e n 1 #(P λ B (M) (x 0,λ β )) N(M) λ ) is an asymptotically unbiased and consistent estimator of Sc (M) x 0.

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