Quenched invariance principle for random walks on Poisson-Delaunay triangulations

Transcription

1 Quenched invariance principle for random walks on Poisson-Delaunay triangulations Institut de Mathématiques de Bourgogne Journées de Probabilités Université de Rouen Jeudi 17 septembre 2015

2 Introduction Result Proof

3 Poisson point processes Homogeneous Poisson point process ξ with intensity λ (PPP) 1. k 2, A 1,..., A k R d disjoint bounded Borel sets, the r.v. #(ξ A 1),..., #(ξ A k ) are independent 2. A B b (R d ), #(ξ A) is Poisson distributed with mean λ Vol(A)

4 Voronoi tiling and Delaunay triangulation Voronoi cell with nucleus x ξ: { } Vor ξ (x) = y R d : y x 2 y x 2, x ξ

5 Voronoi tiling and Delaunay triangulation Voronoi cell with nucleus x ξ: { } Vor ξ (x) = y R d : y x 2 y x 2, x ξ DT(ξ): Delaunay triangulation of ξ

6 Voronoi tiling and Delaunay triangulation Voronoi cell with nucleus x ξ: { } Vor ξ (x) = y R d : y x 2 y x 2, x ξ DT(ξ): Delaunay triangulation of ξ

7 Random walks in random environments in the literature Recurrence Invariance principles Models and transience annealed quenched Percolation cluster [Berger, Biskup; 07], and random [Grimmett et al.; 93] [De Masi et al.; 89] [Biskup, Prescott; 07], conductances in Z d... Complete graph generated by point proc. in R d, [Caputo et al.; 09] [Faggionato et al.; 06] [Caputo et al.; 13], transition probab. with distance Delaunay ([Addario-Berry, Sarkar; 05]) [Ferrari et al.; 12], triangulation (d = 2) generated by PPP

8 Random walks in random environments in the literature Recurrence Invariance principles Models and transience annealed quenched Percolation cluster [Berger, Biskup; 07], and random [Grimmett et al.; 93] [De Masi et al.; 89] [Biskup, Prescott; 07], conductances in Z d... Complete graph generated by point proc. in R d, [Caputo et al.; 09] [Faggionato et al.; 06] [Caputo et al.; 13], transition probab. with distance Delaunay ([Addario-Berry, Sarkar; 05]) [Ferrari et al.; 12], triangulation [R.;?] (d = 2) generated by PPP [R.; 15] [R.; 15], (d 2)

9 Introduction Result Proof

10 Quenched invariance principle ξ distributed according to a PPP in R d, d 2 ( ) X ξ n : simple nearest neighbor random walk on DT(ξ) n N Px ξ : law of ( ) Xn ξ starting at x n N ( Bε ξ (t) = ε X ξ + ( ε 2 t ε 2 t ) ( )) X ξ X ξ, t 0 ε 2 t ε 2 t +1 ε 2 t

11 Quenched invariance principle ξ distributed according to a PPP in R d, d 2 ( ) X ξ n : simple nearest neighbor random walk on DT(ξ) n N Px ξ : law of ( ) Xn ξ starting at x n N ( Bε ξ (t) = ε X ξ + ( ε 2 t ε 2 t ) ( )) X ξ X ξ, t 0 ε 2 t ε 2 t +1 ε 2 t Theorem [R.; 15] For all T > 0, for a.e. ξ, for all x ξ, the law of ( Bε ξ (t) ) induced by 0 t T Pξ x ( on C([0, ) T ]; R d ) converges weakly, as ε 0, to the law of a Brownian motion B ξ t starting at x with covariance matrix σ 2 I d where σ 2 is positive and 0 t T does not depend on ξ.

12 Introduction Result Proof

13 Martingale decomposition For a.e. ξ, for x ξ, we want to write: X ξ n = M ξ n + R ξ n with and ( M ξ n ) n N : Pξ x -martingale converges to a BM by Lindeberg-Feller functional CLT (R ) ξ n : corrector n N negligible at the diffusive scale: Rn ξ lim n n = 0 a.s.

14 Construction of the martingale (1/3) Let µ be the measure on N 0 R d defined by: f d µ = (f (x) f (0)) P 0(d ξ 0 ), N 0 x ξ 00 where P 0 denotes the Palm measure associated to the PPP.

15 Construction of the martingale (1/3) Let µ be the measure on N 0 R d defined by: f d µ = (f (x) f (0)) P 0(d ξ 0 ), N 0 x ξ 00 where P 0 denotes the Palm measure associated to the PPP. Weil decomposition of L 2 (µ) L 2 (µ) = L 2 pot(µ) L 2 sol(µ) with L 2 pot(µ): closure of the space of gradients of bounded meas. functions

16 Construction of the martingale (2/3) Consider the projection p : (ξ 0, x) x.

17 Construction of the martingale (2/3) Consider the projection p : (ξ 0, x) x. Note that it is in L 2 (µ) since p 2 d µ = x 2 P 0(d ξ 0 ) N 0 x ξ 00 deg ξ 0(0) max N x x 2 P 0(d ξ 0 ) 0 ξ 00 ( ) 1 deg ξ 0(0) 2 P 0(d ξ 0 2 ) N 0 max N x 0 x 4 P 0(d ξ 0 ) ξ <.

18 Construction of the martingale (2/3) Consider the projection p : (ξ 0, x) x. Note that it is in L 2 (µ) since p 2 d µ = x 2 P 0(d ξ 0 ) N 0 x ξ 00 deg ξ 0(0) max N x x 2 P 0(d ξ 0 ) 0 ξ 00 ( ) 1 deg ξ 0(0) 2 P 0(d ξ 0 2 ) N 0 max N x 0 x 4 P 0(d ξ 0 ) ξ <. So, we can write p = χ + L 2 pot (µ) ϕ. L 2 sol (µ)

19 Construction of the martingale (3/3) Since ϕ L 2 sol(µ) is antisymmetric ( ) ξ 0, x = 0, for P 0-a.e. ξ 0, ϕ x ξ 00

20 Construction of the martingale (3/3) Since ϕ L 2 sol(µ) is antisymmetric ( ) ξ 0, x = 0, for P 0-a.e. ξ 0, ϕ x ξ 00 and actually ϕ (τ xξ, y x) = 0, for all x ξ, for P-a.e. ξ. y ξ x

21 Construction of the martingale (3/3) Since ϕ L 2 sol(µ) is antisymmetric ( ) ξ 0, x = 0, for P 0-a.e. ξ 0, ϕ x ξ 00 and actually ϕ (τ xξ, y x) = 0, for all x ξ, for P-a.e. ξ. y ξ x Thus, n 1 Mn ξ = is a P ξ x -martingale. i=0 ( ϕ τ X ξ i ) ( ξ, X ξ i+1 X ξ i = ϕ τ X ξ 0 ξ, X ξ n X ξ 0 )

22 The corrector ( Rn ξ = Xn ξ Mn ξ = χ τ X ξ 0 ξ, X ξ n X ξ 0 ) It remains to prove that lim n + max χ(τ xξ, y x) = 0 a.s.. y ξ [ n,n] d n

23 The corrector ( Rn ξ = Xn ξ Mn ξ = χ τ X ξ 0 ξ, X ξ n X ξ 0 ) It remains to prove that lim n + max χ(τ xξ, y x) = 0 a.s.. y ξ [ n,n] d n By the maximum principle, it suffices to show that lim n + χ(τ xξ, y x) max = 0 a.s. y G (ξ) [ n,n] d n where G (ξ) is an infinite connected subgraph of DT(ξ) such that each connected component of DT(ξ) \ G (ξ) is finite.

24 Construction of G (ξ) (1/3) We part R d into boxes B z of side K, z Z d, and subdivise each box into sub-boxes of side α d K. We say that B z is good if: each sub-box of side α d K included in B z = B z contains at least a point of ξ, z z 1 # ( ξ B z ) D. If K and D are well chosen, the process of the good boxes stochastically dominates an indep. percolation process with parameter p (1 p c(z d ), 1).

25 Construction of G (ξ) (2/3) G = the infinite cluster of percolation G L = the maximal connected component of G [ L, L] d

26 Construction of G (ξ) (3/3) G (ξ) = {x ξ : z G s.t. Vor ξ (x) B z } G L (ξ) = {x ξ : z G L s.t. Vor ξ (x) B z }

27 Sublinearity of the corrector in G (ξ) à la [Biskup, Prescott, 07] (1/3) Sublinearity on average: ε > 0, lim n + 1 n d y G (ξ) [ n,n] d 1 χ(τx ξ,y x) εn = 0 ergodicity arguments directional sublinearity extension dimension by dimension

28 Sublinearity of the corrector in G (ξ) à la [Biskup, Prescott, 07] (2/3) Polynomial growth: θ > 0, lim n χ(τ xξ, y x) max = 0 y G (ξ) [ n,n] d n θ analytic properties of χ

29 Sublinearity of the corrector in G (ξ) à la [Biskup, Prescott, 07] (3/3) Diffusive bounds: Define T 1 = inf{j 1 : X ξ j G (ξ)}. The random walk (Y ξ t ) t 0 with generator ( ) L ξ f (y) = Py ξ [ X ξ T 1 = y ]( f (y ) f (y) ) n 1 t n y G (ξ) satisfies [ ] [ ]) sup max sup max (t 2 1 E ξ y Y ξ y G (ξ) [ n,n] d t y, t d 2 P ξ y Y ξ t = y < + p.s. distance comparison isoperimetric inequalities heat kernel estimates for (Y ξ t ) t 0 (see [Morris, Peres; 05])

30 Isoperimetric inequality in G L (ξ) (1/5) For A G L (ξ), define I L,ξ A = x A y A 1 c x y in DT(ξ) deg L (A) where A c = G L (ξ) \ A and deg L (A) = x A deg L (x). Claim There exists c > 0 such that a.s. for L large enough ( ) I L,ξ 1 1 A c min, deg L (A) d 1 log(l) d 1 d for every A G L (ξ) with deg L (A) 1 2 deg L (G L(ξ)).

31 Isoperimetric inequality in G L (ξ) (2/5) For A G L (ξ), define L(A) = {z G L : x A s.t. Vor ξ (x) B z }. Note that #L(A) 2 d deg L (A) #(A) }{{} D#L(A) max deg L (x) x A }{{} D D 2 #L(A). (1)

32 Isoperimetric inequality in G L (ξ) (2/5) For A G L (ξ), define L(A) = {z G L : x A s.t. Vor ξ (x) B z }. Note that #L(A) 2 d deg L (A) #(A) }{{} D#L(A) max deg L (x) x A }{{} D D 2 #L(A). (1) We distinguish the cases whether or not #L(A) > ( d+2 D 2 ) #GL.

33 Isoperimetric inequality in G L (ξ) (3/5): case #L(A) > ( d+2 D 2 ) #GL Since deg L (A) 1 2 deg L (G L(ξ)), we have #L(A c ) #G L 2 d+1 D 2 and # (L(A) L(A c )) #G L 2 d+1 D 2.

34 Isoperimetric inequality in G L (ξ) (3/5): case #L(A) > ( d+2 D 2 ) #GL Since deg L (A) 1 2 deg L (G L(ξ)), we have #L(A c ) #G L 2 d+1 D 2 and # (L(A) L(A c )) #G L 2 d+1 D 2. If z L(A) L(A c ), there exists an edge between a point of A and a point of A c contained in B z = B z. z z 1

35 Isoperimetric inequality in G L (ξ) (3/5): case #L(A) > ( d+2 D 2 ) #GL Since deg L (A) 1 2 deg L (G L(ξ)), we have #L(A c ) #G L 2 d+1 D 2 and # (L(A) L(A c )) #G L 2 d+1 D 2. If z L(A) L(A c ), there exists an edge between a point of A and a point of A c contained in B z = B z. This implies that x A z z 1 y A c 1 x y # (L(A) L(A c )) #G L 3 d 4 6 d D 2 so that I L,ξ A d D 4.

36 Isoperimetric inequality in G L (ξ) (4/5): case #L(A) ( d+2 D 2 ) #GL If z L(A) and z G L \ L(A) are neighbors, there exists an edge between a point of A and a point of A c contained in B z B z. It follows that x A y A c 1 x y δ max (# ( L(A)), # ( (G L \ L(A)))).

37 Isoperimetric inequality in G L (ξ) (4/5): case #L(A) ( d+2 D 2 ) #GL If z L(A) and z G L \ L(A) are neighbors, there exists an edge between a point of A and a point of A c contained in B z B z. It follows that Besides, Hence, and x A y A c 1 x y δ max (# ( L(A)), # ( (G L \ L(A)))). #L(A) ( ) 1 1 (#L(A) + # (G 2 d+2 D 2 L \ L(A))). deg L (A) D 2 #L(A) D 2 (2 d+2 D 2 1)# (G L \ L(A)) I L,ξ A for A = L(A) or G L \ L(A). δ # ( A) D 2 (2 d+2 D 2 1) #A

38 Isoperimetric inequality in G L (ξ) (5/5): case #L(A) ( d+2 D 2 ) #GL By applying Isoperimetric inequality in G L (see e.g. [Caputo, Faggionato; 07]) There exists κ > 0 such that almost surely for L large enough, for A G (L) with 0 < #(A) 1 #(G 2 (L)) { } #( A) #(A) κ min 1 1,. #(A) d 1 log(l) d 1 d and then (1), one finally obtains that ( I L,ξ κδ A D 2 (2 d+2 D 2 1) min 1 1, #(A) d 1 ( κδ 2D 2 (2 d+2 D 2 1) min 1, deg L (A) d 1 log(l) d d 1 1 ) log(l) d d 1 ).

39 Thank you!