Gibbs point processes with geometrical interactions. Models for strongly structured patterns

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1 Gibbs point processes with geometrical interactions. Models for strongly structured patterns David Dereudre, Laboratoire Paul Painlevé, Lille (joint work with R. Drouilhet and H.-O. Georgii) 8th September 2014, Lorentz Center, Leiden Workshop Mathematical Facets of Crystallization

2 1 Hamiltonian with geometrical interaction 2 Existence results 3 Some words about the proof of the Theorem. 4 Open questions

3 1 Hamiltonian with geometrical interaction

4 Classical pairwise interaction Pairwise interaction in a gas. H Λ (ω Λ ) = β Example : Lennard Jones potential ( σ 12 ϕ(x, y) = 4ɛ σ 0, ɛ 0. {x,y} ω Λ ϕ(x, y). σ6 x y 12 x y 6 ),

5 Delaunay interaction Pairwise interaction for Delaunay edges. H Λ (ω Λ ) = β ϕ(x, y). Example : Quadratic interaction Fibre interaction {x,y} ω Λ {x,y}delaunay edge ϕ(x, y) = x y 2 ϕ(x, y) = 1I [1,2] ( x y )

6 Voronoi interaction H Λ (ω Λ ) = β ϕ(vor(x, ω Λ )). x ω Λ Example : Perimeter penalization ϕ(vor(x, ω Λ )) = perimeter(vor(x, ω Λ ).

7 Inhibition geometric interaction H Λ (ω Λ ) = + 1I φ(object) K. object O(ω Λ ) Example : Inhibition for the Voronoi cell geometry ϕ(vor(x, ω Λ )) = Volume(Vor(x, ω Λ ). ϕ(vor(x, ω Λ )) = Diameter(Vor(x, ω Λ ). ϕ(vor(x, ω Λ )) = N f (Vor(x, ω Λ ).

8 General formalism for the interaction A hyper-graph structure is a measurable subset E in Ω f Ω such that η ω for any (η, ω) E. If (η, ω) E, we said that η is a hyper-edge of ω and E(ω) denotes the set of hyper-edges of ω.

9 General formalism for the interaction A hyper-graph structure is a measurable subset E in Ω f Ω such that η ω for any (η, ω) E. If (η, ω) E, we said that η is a hyper-edge of ω and E(ω) denotes the set of hyper-edges of ω. A hyper-potential is a measurable function ϕ from E to R {+ }.

10 General formalism for the interaction A hyper-graph structure is a measurable subset E in Ω f Ω such that η ω for any (η, ω) E. If (η, ω) E, we said that η is a hyper-edge of ω and E(ω) denotes the set of hyper-edges of ω. A hyper-potential is a measurable function ϕ from E to R {+ }. We assume that ϕ satises the nite horizon property : for all (η, ω) E there exits a bounded set η,ω such that (η, ω) E and ϕ(η, ω) = ϕ(η, ω) as soon as ω η,ω = ω η,ω.

11 General formalism for the interaction A hyper-graph structure is a measurable subset E in Ω f Ω such that η ω for any (η, ω) E. If (η, ω) E, we said that η is a hyper-edge of ω and E(ω) denotes the set of hyper-edges of ω. A hyper-potential is a measurable function ϕ from E to R {+ }. We assume that ϕ satises the nite horizon property : for all (η, ω) E there exits a bounded set η,ω such that (η, ω) E and ϕ(η, ω) = ϕ(η, ω) as soon as ω η,ω = ω η,ω. We assume that E and ϕ are stationary.

12 Examples of hyper-edge and horizon ω Let ω be a conguration

13 Examples of hyper-edge and horizon ω η Let ω be a conguration η ω is a hyper-edge if (η, ω) E

14 Examples of hyper-edge and horizon ω η η,ω Let ω be a conguration η ω is a hyper-edge if (η, ω) E ϕ(η, ω) = ϕ(η, ω η,ω ω c ) η,ω

15 For the Delaunay triangles ω Let ω be a conguration

16 For the Delaunay triangles ω η Let ω be a conguration η ω is a Delaunay triangle

17 For the Delaunay triangles ω η η,ω Let ω be a conguration η ω is a Delaunay triangle ϕ(η, ω) = ϕ(η)

18 For the Voronoi cell ω x x,ω x ω is a center of a Voronoi cell ϕ(x, ω) = ϕ(x, ω x,ω )

19 Hamiltonian Let ω be a conguration and Λ be a bounded set in R 2, the energy of ω inside Λ is dened by H Λ (ω) = ϕ(η, ω). η E(ω) η,ω Λ

20 Hamiltonian Let ω be a conguration and Λ be a bounded set in R 2, the energy of ω inside Λ is dened by H Λ (ω) = ϕ(η, ω). η E(ω) η,ω Λ ω is said admissible if for every Λ and every conguration ω Λ the Hamiltonian H Λ (ω Λ c ω Λ ) is well-dened and if Z Λ (ω Λ c) := e H Λ(ω Λ c ω Λ ) πλ(dω z Λ) is nite and positive ; 0 < Z Λ (ω Λ c) < +. Ω z denotes the set of admissible congurations for (z, µ).

21 Gibbs measures Denition A probability measure P on Ω is a Gibbs measure for the intensity z > 0, the law of marks µ and the hyper-potential ϕ if P (Ω z) = 1 and if for every bounded set Λ and P -almost every ω Λ c P (dω Λ ω Λ c) = 1 Z Λ (ω Λ c) e H Λ(ω Λc ω Λ ) π z Λ(dω Λ ).

22 2 Existence results

23 State of the art - Classical Statistical Mechanics : pairwise interaction, multibody interaction, hardcore interaction. Ruelle (1970), superstable and lower regular interactions. Preston (1976), general Theorems

24 State of the art - Classical Statistical Mechanics : pairwise interaction, multibody interaction, hardcore interaction. Ruelle (1970), superstable and lower regular interactions. Preston (1976), general Theorems - Geometrical interaction : Baddeley, Møller (1989), nearest-neighbour interaction ISR Bertin, Billiot, Drouilhet (1999), innite volume Gibbs measure for nearest-neighbour interaction, AAP Der., (2008) Gibbs Delaunay tessellation with Geometric Hardcore Conditions, JSP Der., Drouilhet and Georgii, (2012) Existence of Gibbsian point processes with geometry-dependent interactions, PTRF

25 Range of the interaction Denition ( Range assumption (R)) l R, n R N, R < such that For all (n, ω) in E, there exists a horizon η,ω satisfying : For all x, y η,ω, there exist l open balls B 1,..., B l (with l l R ) such that the set l i=1 B i is connected and contains x and y, for every i, radius (B i ) R or N Bi (ω) n R.

26 Stability Inuence set : Λ (ω) = η,ωλ c ω Λ. ω Λ Ω Λ η E(ω Λ c ω Λ ) η,ωλ c ω Λ Λ

27 Stability Inuence set : Λ (ω) = η,ωλ c ω Λ. ω Λ Ω Λ η E(ω Λ c ω Λ ) η,ωλ c ω Λ Λ Λ (ω) is a maximal set around Λ for which the points of ω in this set impact on the value of H Λ (ω Λ c.).

28 Stability Inuence set : Λ (ω) = η,ωλ c ω Λ. ω Λ Ω Λ η E(ω Λ c ω Λ ) η,ωλ c ω Λ Λ Λ (ω) is a maximal set around Λ for which the points of ω in this set impact on the value of H Λ (ω Λ c.). Denition ( Stability assumption (S)) c S 0 such that for all ω and Λ H Λ (ω) c S #(ω Λ (ω)).

29 Upper bound assumption Let C be a parallelogram in R 2 and (C(k)) k Z d the associated paving of R 2. Let Γ be an event of Ω C and Γ = {ω Ω : ω C(k) Γ k for all k Z d} Denition (Upper bound assumption (U)) We can choose C and Γ such that Uniform connement : r > 0 such that for all ω Γ, Λ ( ω) Λ B(0, r). Uniform summability : c + G := sup ω Γ η E( ω): η C ϕ + (η, ω) #(ˆη) <, where ˆη := {k Z d : η C(k) }. Non-rigidity : e z C π z C (Γ) > ec G.

30 Existence Theorem Theoreme (Der.-Drouihlet-Georgii) For every hyper-graph structure E, every hyper-potential ϕ and every activity z > 0 satisfying (R), (S) and (U), there exists at least one stationary Gibbs measure P.

31 Existence Theorem Theoreme (Der.-Drouihlet-Georgii) For every hyper-graph structure E, every hyper-potential ϕ and every activity z > 0 satisfying (R), (S) and (U), there exists at least one stationary Gibbs measure P. Remarks : - No superstability assumption is required

32 Existence Theorem Theoreme (Der.-Drouihlet-Georgii) For every hyper-graph structure E, every hyper-potential ϕ and every activity z > 0 satisfying (R), (S) and (U), there exists at least one stationary Gibbs measure P. Remarks : - No superstability assumption is required - Stability and uniform nite range existence.

33 Existence Theorem Theoreme (Der.-Drouihlet-Georgii) For every hyper-graph structure E, every hyper-potential ϕ and every activity z > 0 satisfying (R), (S) and (U), there exists at least one stationary Gibbs measure P. Remarks : - No superstability assumption is required - Stability and uniform nite range existence. - ϕ(η, ) may tend to innity when the size of η tends to innity.

34 Existence Theorem Theoreme (Der.-Drouihlet-Georgii) For every hyper-graph structure E, every hyper-potential ϕ and every activity z > 0 satisfying (R), (S) and (U), there exists at least one stationary Gibbs measure P. Remarks : - No superstability assumption is required - Stability and uniform nite range existence. - ϕ(η, ) may tend to innity when the size of η tends to innity. - We cover a large range of applications : Classical pairwise (and multibody) interaction, Delaunay-Voronoi interaction, etc

35 Voronoi interaction Example : Perimeter penalization interaction ϕ(x, ω) = perimeter(vor(x, ω)).

36 Voronoi interaction Example : Perimeter penalization interaction ϕ(x, ω) = perimeter(vor(x, ω)). The gibbs measures exist only for z large enough. It is due to the Non-rigidity assumption e z C π z C(Γ) > e c G e z C e z C z A > e c G z A > e c G

37 Voronoi interaction Example : Perimeter penalization interaction ϕ(x, ω) = perimeter(vor(x, ω)). The gibbs measures exist only for z large enough. It is due to the Non-rigidity assumption e z C π z C(Γ) > e c G e z C e z C z A > e c G z A > e c G Heuristic : for z small, the size and the perimeter of cells are large. The interaction blows up when z 0.

38 Voronoi interaction Example : Perimeter penalization interaction ϕ(x, ω) = perimeter(vor(x, ω)). The gibbs measures exist only for z large enough. It is due to the Non-rigidity assumption e z C π z C(Γ) > e c G e z C e z C z A > e c G z A > e c G Heuristic : for z small, the size and the perimeter of cells are large. The interaction blows up when z 0. Conjecture : The Gibbs measures do not exist for z small enough.

39 3 Some words about the proof of the Theorem.

40 Stationary nite volume Gibbs measures Λ n denotes the cube k { n,...,n} d C(k).

41 Stationary nite volume Gibbs measures Λ n denotes the cube k { n,...,n} d C(k). Let ω be a pseudo-periodic conguration in Γ. P n (dω Λn ) = 1 Z n ( ω) e H Λn (ω Λn ω Λc n ) π z Λ n (dω Λn ).

42 Stationary nite volume Gibbs measures Λ n denotes the cube k { n,...,n} d C(k). Let ω be a pseudo-periodic conguration in Γ. P n (dω Λn ) = 1 Z n ( ω) e H Λn (ω Λn ω Λc n ) π z Λ n (dω Λn ). ˆP n is the probability measure such that the congurations in the disjoint blocks Λ n + (2n+1)k, for k Z d, are independent with identical distribution P n.

43 Stationary nite volume Gibbs measures Λ n denotes the cube k { n,...,n} d C(k). Let ω be a pseudo-periodic conguration in Γ. P n (dω Λn ) = 1 Z n ( ω) e H Λn (ω Λn ω Λc n ) π z Λ n (dω Λn ). ˆP n is the probability measure such that the congurations in the disjoint blocks Λ n + (2n+1)k, for k Z d, are independent with identical distribution P n. The stationary nite volume Gibbs measure is P n = Λ n Λ 1 ˆPn τx 1 dx. n

44 Tightness tool Let P be a stationary probability measure on Ω.

45 Tightness tool Let P be a stationary probability measure on Ω. The relative entropy of P Λ with respect to πλ z is dened by = f, if dp Λ dπ z Λ { f ln f dπ I(P Λ πλ z z ) = Λ + otherwise.

46 Tightness tool Let P be a stationary probability measure on Ω. The relative entropy of P Λ with respect to πλ z is dened by = f, if dp Λ dπ z Λ { f ln f dπ I(P Λ πλ z z ) = Λ + otherwise. The specic entropy of P with respect to π z is dened by I z (P ) = lim Λ R 2 Λ 1 I(P Λ π z Λ).

47 Tightness tool Let P be a stationary probability measure on Ω. The relative entropy of P Λ with respect to πλ z is dened by = f, if dp Λ dπ z Λ { f ln f dπ I(P Λ πλ z z ) = Λ + otherwise. The specic entropy of P with respect to π z is dened by I z (P ) = lim Λ R 2 Λ 1 I(P Λ π z Λ). Proposition (Georgii-Zessin (1993)) For all c 1, c 2 0 and z > 0, the set { P P : I z (P ) c 1 z(p ) c 2 } is relatively compact for the local topology.

48 Connement tool Proposition Under assumption (R), for all Λ, there exist (Ω n ) n n0 - Ω n is F Λ B(0,n)\Λ -mesurable - For all ω in Ω n, Λ (ω) Λ B(0, n) - For all P P such that P (ω = ) = 0, P ( n n0 Ω n ) = 1. such that

49 Connement tool Proposition Under assumption (R), for all Λ, there exist (Ω n ) n n0 - Ω n is F Λ B(0,n)\Λ -mesurable - For all ω in Ω n, Λ (ω) Λ B(0, n) - For all P P such that P (ω = ) = 0, P ( n n0 Ω n ) = 1. Thanks to the assumption (U), we prove that the limiting probability measure P satises P (ω = ) < 1. such that

50 Connement tool Proposition Under assumption (R), for all Λ, there exist (Ω n ) n n0 - Ω n is F Λ B(0,n)\Λ -mesurable - For all ω in Ω n, Λ (ω) Λ B(0, n) - For all P P such that P (ω = ) = 0, P ( n n0 Ω n ) = 1. Thanks to the assumption (U), we prove that the limiting probability measure P satises P (ω = ) < 1. The Gibbs measure is P (. ω ). such that

51 4 Open questions

52 Open questions Conjecture : For the perimeter penalization interaction, the Gibbs measures do not exist for z small enough.

53 Open questions Conjecture : For the perimeter penalization interaction, the Gibbs measures do not exist for z small enough. Phase transition results for a well chosen geometrical interaction. Try to use geometric properties of the well structured pattern for proving long dependence in the eld. For example, the inhibition interaction gives, with probability one, good geometric properties.

54 Open questions Conjecture : For the perimeter penalization interaction, the Gibbs measures do not exist for z small enough. Phase transition results for a well chosen geometrical interaction. Try to use geometric properties of the well structured pattern for proving long dependence in the eld. For example, the inhibition interaction gives, with probability one, good geometric properties. It is a large new class of processes. There are a lot of things to do...

Gibbs Voronoi tessellations: Modeling, Simulation, Estimation.

Gibbs Voronoi tessellations: Modeling, Simulation, Estimation. Frédéric Lavancier, Laboratoire Jean Leray, Nantes (France) Work with D. Dereudre (LAMAV, Valenciennes, France). SPA, Berlin, July 27-31,