Introduction
Result
Quenched invariance principle
for random walks
on Poisson-Delaunay triangulations
Arnaud Rousselle
Institut de Mathématiques de Bourgogne
Journées de Probabilités
Université de Rouen
Jeudi 17 septembre 2015
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Introduction
Result
Proof
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Poisson point processes
Homogeneous Poisson point process ξ with intensity λ (PPP)
1. ∀k ≥ 2, ∀A1 , . . . , Ak ⊂ Rd disjoint bounded Borel sets, the r.v.
#(ξ ∩ A1 ), . . . , #(ξ ∩ Ak ) are independent
2. ∀A ∈ Bb (Rd ), #(ξ ∩ A) is Poisson distributed with mean λ · Vol(A)
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Voronoi tiling and Delaunay triangulation
I
Voronoi cell with nucleus x ∈ ξ:
n
o
Vorξ (x) = y ∈ Rd : |y − x|2 ≤ |y − x 0 |2 , ∀x 0 ∈ ξ
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Voronoi tiling and Delaunay triangulation
I
Voronoi cell with nucleus x ∈ ξ:
n
o
Vorξ (x) = y ∈ Rd : |y − x|2 ≤ |y − x 0 |2 , ∀x 0 ∈ ξ
I
DT(ξ): Delaunay triangulation of ξ
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Voronoi tiling and Delaunay triangulation
I
Voronoi cell with nucleus x ∈ ξ:
n
o
Vorξ (x) = y ∈ Rd : |y − x|2 ≤ |y − x 0 |2 , ∀x 0 ∈ ξ
I
DT(ξ): Delaunay triangulation of ξ
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Random walks in random environments in the literature
Models
Recurrence
and transience
Invariance principles
annealed
quenched
Percolation cluster
and random
conductances in Zd
[Grimmett et al.; ’93]
[De Masi et al.; ’89]
[Berger, Biskup; ’07],
[Biskup, Prescott; ’07],
...
Complete graph
generated by
point proc. in Rd ,
transition probab.
& with distance
[Caputo et al.; ’09]
[Faggionato et al.; ’06]
[Caputo et al.; ’13],
Delaunay
triangulation
generated by PPP
([Addario-Berry, Sarkar; ’05])
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
[Ferrari et al.; ’12],
(d = 2)
Introduction
Result
Proof
Random walks in random environments in the literature
Models
Recurrence
and transience
Invariance principles
annealed
quenched
Percolation cluster
and random
conductances in Zd
[Grimmett et al.; ’93]
[De Masi et al.; ’89]
[Berger, Biskup; ’07],
[Biskup, Prescott; ’07],
...
Complete graph
generated by
point proc. in Rd ,
transition probab.
& with distance
[Caputo et al.; ’09]
[Faggionato et al.; ’06]
[Caputo et al.; ’13],
[R.; ?]
[Ferrari et al.; ’12],
(d = 2)
[R.; ’15], (d ≥ 2)
Delaunay
triangulation
generated by PPP
([Addario-Berry, Sarkar; ’05])
[R.; ’15]
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Introduction
Result
Proof
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Quenched invariance principle
I
I
I
I
ξ distributed according to a PPP in Rd , d ≥ 2
Xnξ n∈N : simple nearest neighbor random walk on DT(ξ)
Pxξ : law of Xnξ n∈N starting at x
ξ
ξ
ξ
−2
,
Bεξ (t) = ε Xbε
t − bε−2 tc Xbε
−2 tc + ε
−2 tc+1 − Xbε−2 tc
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
t≥0
Introduction
Result
Proof
Quenched invariance principle
I
I
I
I
ξ distributed according to a PPP in Rd , d ≥ 2
Xnξ n∈N : simple nearest neighbor random walk on DT(ξ)
Pxξ : law of Xnξ n∈N starting at x
ξ
ξ
ξ
−2
,
Bεξ (t) = ε Xbε
t − bε−2 tc Xbε
−2 tc + ε
−2 tc+1 − Xbε−2 tc
t≥0
Theorem [R.; ’15]
For all T > 0, for a.e. ξ, for all x ∈ ξ, the law of Bεξ (t)
0≤t≤T
induced by Pxξ
d
on C([0,
T ]; R ) converges weakly, as ε → 0, to the law of a Brownian motion
ξ
Bt
starting at x with covariance matrix σ 2 Id where σ 2 is positive and
0≤t≤T
does not depend on ξ.
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Introduction
Result
Proof
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Martingale decomposition
For a.e. ξ, for x ∈ ξ, we want to write:
Xnξ = Mnξ + Rnξ
with
Mnξ
Rnξ
n∈N
: Pxξ -martingale
,→ converges to a BM by Lindeberg-Feller functional CLT
and
n∈N
: corrector
Rnξ
= 0 a.s.
n→∞ n
,→ negligible at the diffusive scale: lim
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Construction of the martingale (1/3)
Let µ be the measure on N0 × Rd defined by:
Z
Z X
f dµ =
(f (x) − f (0)) P0 (d ξ 0 ),
N0 x ∼ 0
ξ0
where P0 denotes the Palm measure associated to the PPP.
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Construction of the martingale (1/3)
Let µ be the measure on N0 × Rd defined by:
Z
Z X
f dµ =
(f (x) − f (0)) P0 (d ξ 0 ),
N0 x ∼ 0
ξ0
where P0 denotes the Palm measure associated to the PPP.
Weil decomposition of L2 (µ)
⊥
L2 (µ) = L2pot (µ) ⊕ L2sol (µ)
with
L2pot (µ): closure of the space of gradients of bounded meas. functions
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Construction of the martingale (2/3)
Consider the projection
p : (ξ 0 , x) 7−→ x.
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Construction of the martingale (2/3)
Consider the projection
p : (ξ 0 , x) 7−→ x.
Note that it is in L2 (µ) since
Z
Z X
2
|p| d µ =
|x|2 P0 (d ξ 0 )
N0 x ∼ 0
ξ0
Z
degξ0 (0) max |x|2 P0 (d ξ 0 )
≤
N0
x ∼0
ξ0

1 Z
2

degξ0 (0) P0 (d ξ )
Z
2
≤
0
N0
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
1
2
4
0
max |x| P0 (d ξ ) < ∞.
N0 x ξ∼0 0
Introduction
Result
Proof
Construction of the martingale (2/3)
Consider the projection
p : (ξ 0 , x) 7−→ x.
Note that it is in L2 (µ) since
Z
Z X
2
|p| d µ =
|x|2 P0 (d ξ 0 )
N0 x ∼ 0
ξ0
Z
degξ0 (0) max |x|2 P0 (d ξ 0 )
≤
N0
x ∼0
ξ0

1 Z
2

degξ0 (0) P0 (d ξ )
Z
2
≤
0
1
2
4
max |x| P0 (d ξ ) < ∞.
N0 x ξ∼0 0
N0
So, we can write
p=
χ
∈L2pot (µ)
+
ϕ
∈L2sol (µ)
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
0
.
Introduction
Result
Proof
Construction of the martingale (3/3)
Since ϕ ∈ L2sol (µ) is antisymmetric
X 0 ϕ ξ , x = 0,
x ∼0
ξ0
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
for P0 -a.e. ξ 0 ,
Introduction
Result
Proof
Construction of the martingale (3/3)
Since ϕ ∈ L2sol (µ) is antisymmetric
X 0 ϕ ξ , x = 0,
for P0 -a.e. ξ 0 ,
x ∼0
ξ0
and actually
X
ϕ (τx ξ, y − x) = 0,
for all x ∈ ξ, for P-a.e. ξ.
y ∼x
ξ
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Proof
Construction of the martingale (3/3)
Since ϕ ∈ L2sol (µ) is antisymmetric
X 0 ϕ ξ , x = 0,
for P0 -a.e. ξ 0 ,
x ∼0
ξ0
and actually
X
ϕ (τx ξ, y − x) = 0,
for all x ∈ ξ, for P-a.e. ξ.
y ∼x
ξ
Thus,
Mnξ =
n−1
X
i=0
ξ
ϕ τX ξ ξ, Xi+1
− Xiξ = ϕ τX ξ ξ, Xnξ − X0ξ
i
is a Pxξ -martingale.
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
0
Introduction
Result
Proof
The corrector
Rnξ = Xnξ − Mnξ = χ τX ξ ξ, Xnξ − X0ξ
0
It remains to prove that
lim
max
n→+∞ y ∈ξ∩[−n,n]d
|χ(τx ξ, y − x)|
= 0 a.s..
n
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Proof
The corrector
Rnξ = Xnξ − Mnξ = χ τX ξ ξ, Xnξ − X0ξ
0
It remains to prove that
lim
max
n→+∞ y ∈ξ∩[−n,n]d
|χ(τx ξ, y − x)|
= 0 a.s..
n
By the maximum principle, it suffices to show that
lim
max
n→+∞ y ∈G∞ (ξ)∩[−n,n]d
|χ(τx ξ, y − x)|
= 0 a.s.
n
where G∞ (ξ) is an infinite connected subgraph of DT(ξ) such that each
connected component of DT(ξ) \ G∞ (ξ) is finite.
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Proof
Construction of G∞ (ξ) (1/3)
We part Rd into boxes Bz of side K , z ∈ Zd , and
subdivise each box into sub-boxes of side αd K .
We say that Bz is good if:
I
each sub-box
of side αd K included in
S
Bz =
Bz0 contains at least a point of ξ,
|z0 −z|≤1
I
# ξ ∩ Bz ≤ D.
If K and D are well chosen, the process of the good boxes stochastically
dominates an indep. percolation process with parameter p ∈ (1 − pc (Zd ), 1).
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Construction of G∞ (ξ) (2/3)
G∞ = ‘the infinite cluster of percolation’
GL = ‘the maximal connected component of G∞ ∩ [−L, L]d ’
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Construction of G∞ (ξ) (3/3)
G∞ (ξ) = {x ∈ ξ : ∃z ∈ G∞ s.t. Vorξ (x) ∩ Bz 6= ∅}
GL (ξ) = {x ∈ ξ : ∃z ∈ GL s.t. Vorξ (x) ∩ Bz 6= ∅}
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Sublinearity of the corrector in G∞ (ξ) ‘à la [Biskup, Prescott, ’07]’ (1/3)
Sublinearity on average:
∀ε > 0, lim
n→+∞
1
nd
X
y ∈G∞ (ξ)∩[−n,n]d
I
ergodicity arguments
I
directional sublinearity
I
extension dimension by dimension
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
1|χ(τx ξ,y −x)|≥εn = 0
Introduction
Result
Proof
Sublinearity of the corrector in G∞ (ξ) ‘à la [Biskup, Prescott, ’07]’ (2/3)
Polynomial growth:
∃θ > 0, lim
max
n→∞ y ∈G∞ (ξ)∩[−n,n]d
I
analytic properties of χ
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
|χ(τx ξ, y − x)|
=0
nθ
Introduction
Result
Sublinearity of the corrector in G∞ (ξ) ‘à la [Biskup, Prescott, ’07]’ (3/3)
Diffusive bounds:
Define T1 = inf{j ≥ 1 : Xjξ ∈ G∞ (ξ)}.
The random walk (Ytξ )t≥0 with generator
X
Lξ f (y ) =
Pyξ XTξ1 = y 0 f (y 0 ) − f (y )
y 0 ∈G∞ (ξ)
satisfies
sup
max
h
i
h
i
1
d
sup max t − 2 Eyξ |Ytξ − y | , t − 2 Pyξ Ytξ = y < +∞ p.s.
n≥1 y ∈G∞ (ξ)∩[−n,n]d t≥n
I
distance comparison
I
isoperimetric inequalities
I
heat kernel estimates for (Ytξ )t≥0 (see [Morris, Peres; ’05])
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (1/5)
For A ⊂ GL (ξ), define
P
IAL,ξ
=
x∈A
P
y ∈Ac
1x∼y
in DT(ξ)
degL (A)
P
where Ac = GL (ξ) \ A and degL (A) = x∈A degL (x).
Claim
There exists c > 0 such that a.s. for L large enough
IAL,ξ
≥ c min
1
1
1 ,
d
degL (A) d log(L) d−1
for every A ⊂ GL (ξ) with degL (A) ≤
1
2
degL (GL (ξ)).
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
!
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (2/5)
For A ⊂ GL (ξ), define
L(A) = {z ∈ GL : ∃x ∈ A s.t. Vorξ (x) ∩ Bz 6= ∅} .
Note that
#L(A)
≤ degL (A) ≤ #(A) max degL (x) ≤ D 2 #L(A).
x∈A
2d
| {z } |
{z
}
≤D#L(A)
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
≤D
(1)
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (2/5)
For A ⊂ GL (ξ), define
L(A) = {z ∈ GL : ∃x ∈ A s.t. Vorξ (x) ∩ Bz 6= ∅} .
Note that
#L(A)
≤ degL (A) ≤ #(A) max degL (x) ≤ D 2 #L(A).
x∈A
2d
| {z } |
{z
}
≤D#L(A)
≤D
We distinguish the cases whether or not #L(A) > 1 −
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
1
2d+2 D 2
#GL .
(1)
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (3/5): case #L(A) > 1 −
Since degL (A) ≤
1
2
1
2d+2 D 2
degL (GL (ξ)), we have
#L(Ac ) ≥
#GL
2d+1 D 2
and
# (L(A) ∩ L(Ac )) ≥
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
#GL
.
2d+1 D 2
#GL
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (3/5): case #L(A) > 1 −
Since degL (A) ≤
1
2
1
2d+2 D 2
#GL
degL (GL (ξ)), we have
#L(Ac ) ≥
#GL
2d+1 D 2
and
# (L(A) ∩ L(Ac )) ≥
#GL
.
2d+1 D 2
If z ∈ L(A) ∩ L(Ac ), there
S exists an edge between a point of A and a point of
Ac contained in Bz =
Bz0 .
|z0 −z|≤1
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (3/5): case #L(A) > 1 −
Since degL (A) ≤
1
2
1
2d+2 D 2
#GL
degL (GL (ξ)), we have
#L(Ac ) ≥
#GL
2d+1 D 2
and
# (L(A) ∩ L(Ac )) ≥
#GL
.
2d+1 D 2
If z ∈ L(A) ∩ L(Ac ), there
S exists an edge between a point of A and a point of
Ac contained in Bz =
Bz0 .
|z0 −z|≤1
This implies that
XX
x∈A y ∈Ac
1x∼y ≥
# (L(A) ∩ L(Ac ))
#GL
≥
3d
4 × 6d × D 2
so that
IAL,ξ ≥
1
.
4 × 6d × D 4
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Isoperimetric inequality in GL (ξ) (4/5): case #L(A) ≤ 1 −
Proof
1
2d+2 D 2
#GL
If z ∈ L(A) and z0 ∈ GL \ L(A) are neighbors, there exists an edge between a
point of A and a point of Ac contained in Bz ∪ Bz0 .
It follows that
XX
1x∼y ≥ δ max (# (∂L(A)) , # (∂ (GL \ L(A)))) .
x∈A y ∈Ac
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Isoperimetric inequality in GL (ξ) (4/5): case #L(A) ≤ 1 −
Proof
1
2d+2 D 2
#GL
If z ∈ L(A) and z0 ∈ GL \ L(A) are neighbors, there exists an edge between a
point of A and a point of Ac contained in Bz ∪ Bz0 .
It follows that
XX
1x∼y ≥ δ max (# (∂L(A)) , # (∂ (GL \ L(A)))) .
x∈A y ∈Ac
Besides,
#L(A) ≤
1−
1
2d+2 D 2
(#L(A) + # (GL \ L(A))) .
Hence,
degL (A) ≤ D 2 #L(A) ≤ D 2 (2d+2 D 2 − 1)# (GL \ L(A))
and
IAL,ξ ≥
# (∂A)
δ
D 2 (2d+2 D 2 − 1) #A
for A = L(A) or GL \ L(A).
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Introduction
Result
Proof
Isoperimetric inequality in GL (ξ) (5/5): case #L(A) ≤ 1 −
1
2d+2 D 2
#GL
By applying
Isoperimetric inequality in GL (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) ≤ 21 #(G(L) )
(
)
#(∂A)
1
1
≥ κ min
,
.
1
d
#(A)
#(A) d log(L) d−1
and then (1), one finally obtains that
IAL,ξ ≥
κδ
min
D 2 (2d+2 D 2 − 1)
κδ
≥
min
2D 2 (2d+2 D 2 − 1)
1
1
,
!
1
d
#(A) d log(L) d−1
1
1
1 ,
d
degL (A) d log(L) d−1
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
!
.
Introduction
Result
Thank you!
Arnaud Rousselle
Quenched invariance principle for random walks on Poisson-Delaunay triangulations
Proof