• Aucun résultat trouvé

# The 2d-Directed Spanning Forest is almost surely a tree

N/A
N/A
Protected

Academic year: 2021

Partager "The 2d-Directed Spanning Forest is almost surely a tree"

Copied!
15
0
0

Texte intégral

(1)

HAL

HAL, est

(2)

## The 2d-Directed Spanning Forest is almost surely a tree

### David Coupier, Viet Chi Tran June 27, 2011

Abstract

We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no

bi-infinite paths in the DSF. 2

R2

R2

x

y

x

y

R2

x

Theorem 1.

x

R2

x

(3)

### (a) (b)

0 20 40 60 80 100

020406080100

x

y

### Figure 1:

Simulations of the Directed Spanning Forest. The paths with direction ex and coming from vertices with abscissa 0 ≤x≤5 and ordinates 0 ≤y ≤100and their ancestors are represented in bold red lines.

Z2

Z2

R2

Z2

π2

x

R2

(4)

Y Y

(x, y)

### Figure 2:

Y,Y and(x, y)are points of the PPP N. In the DSF with direction ex,Y andY are some descendants of (x, y) and Y is the ancestor of Y. The ancestor of (x, y) cannot belong to the hatched region.

x

x

R2

X

x

R2

′′

R2

(5)

′′

X

Y

R2

P

P

P

P

N

N

m,M

m,M

X

m,M

Y

m,M

m,M

m,M

m,M

X

Y

Lemma 2.

P

P

m,M

N

L,m,M

2

L,m,M

m,Mz

m,M

L,m,M

m,Mz

m,MO

m,M

m,Mz

m,Mz

L,m,M

m,Mz

m,Mz

m,Mz

m,Mz

m,Mz

L,m,M

L,m,M

E

L,m,M

E

 X

z∈ZL,m,M

1Fz

m,M

2P

m,M

P

m,M

E

L,m,M

2

E

L,m,M

3/2

L,m,M

Lemma 3.

N

N

E

L,m,M

3/2

(6)

L,m,M

L,m,M

L<

>L

L<

>L

L,m,M

3/2

<L

L,m,M

L

L,m,M

L<

L

L

3/2

>L

L,m,M

L,m,M

L,m,M

E

E

>L

m,M

P

m,M

Y

m,M

Y

m,M

m,M

m,M

m,M

N

δm,M

m−δ,M

X

Y

X

Y

m,M

m,M

m,M

m,M

δm,M

X

Y

m−δ,M

m,M

m,M

m,M

m−δ,M

m,M

Lemma 4.

P

N

P

δm,M

(7)

P

P

N

m,M

X

Y

X

m,M∈N

P

m,M

X

Y

N

0

1

2

x

k

k≥0

0

θk

θk+1

θk+1−1

θk+1

k>0

X

1

2

P

m,M

X

Y

R→+∞

P

m,M

X

Y

X

Y

m+δ,R

m+δ,R

!

N

m,Mδ

X

Y

Z

m−δ,M

m,M

m,M

Lemma 5.

P

N

P

m,Mδ

X

Y

Z

m,Mδ

X

Y

Z

Z

δ,ℓm,M

δ,0m,M

δm,M

m,M

X

Y

m−δ,M

m,M

Z

P

δ,ℓm,M

δ,km,M

P

 [

−n≤ℓ≤n

δ,ℓm,M

X

−n≤ℓ≤n

P

δ,ℓm,M

P

δm,M

P

δm,M

P

δ,ℓm,M

δ,km,M

Z

(8)

m−δ,M

m−δ,M

m,M

m,M

1

2

3

4

1

3

2

δ,ℓm,M

δ,km,M

1

4

x= 0

γ1

γ2andγ3 γ4

### Figure 3:

Here is a representation of the fourth paths γ1, γ2, γ3, γ4 corresponding to the event Am,MAkm,M. On the axisx= 0, the two rectangles are the cells(0,2M ℓ) +Cm,M and(0,2M k) +Cm,M. While the pathsγ2 andγ3 coalesce, it remains three paths amongγ1, γ2, γ3, γ4 which are disjoint.

m,M

2

m,M

m,M

3

4

1

3

2

4

P

m,Mδ

P

m,M

m,Mδ

m,M

m,Mδ

m,Mδ

m,Mδ

Rm,M

Rm,M

m,M

(9)

Rm,M

0

0

1

R2

k+1

k+1

k

k+1

k

k

ikα

n+1

n+1

−1

−n

−n−1

k

Rm,M

m,M

γY

γX

γZ

### Figure 4:

Here are the setΛRm,M and the Directed Spanning Forest constructed on the PPP N satisfying the eventDR,εBm,Mδ . The disjoint infinite pathsγX, γY, γZ of the eventBδm,M are represented in bold.

The pointsX, Y, ZCmδ,M are in gray. The hatched area is the strip of widthδbetweenEmδ,M and Em,M that the pathsγX, γY andγZ have to cross.

R,ε

k

Rm,M

(10)

k

k

in

out

Rm,M

R2

Rm,M

in

out

P

R,ε

in

P

R,ε

in

out

P

R,ε

R,ε

in

out

Rm,M

in

out

in

X

Y

Z

m,Mδ

in

out

m,Mδ

in

out

m,Mδ

out

P

m,Mδ

out

P

m,Mδ

in

out

P

m,Mδ

m,Mδ

out

m,Mδ

in

out

in

m,M

R,ε

in

p

2

2

Lemma 6.

m,M

m,M

p

2

2

R,ε

in

m,Mδ

out

R,ε

in

out

m,Mδ

in

out

P

R,ε

m,Mδ

P

R,ε

in

out

m,Mδ

in

out

P

R,ε

in

m,Mδ

out

P

R,ε

in

P

m,Mδ

out

R,ε

m,Mδ

m,M

X

Y

Z

δm,M

X

k

k+1

k

k+1

m,M

−k

−k−1

n+1

n+1

X

n+1

Y

Z

−n−1

X

R,ε

m,Mδ

A

m,M

X

X

X

Y

Z

(11)

m,M

p

2

2

m,M

m,M

m,M

q

2

2

p

2

2

q

2

2

[

X∈Q

x

c

c

R2

Theorem 7.

c

c

N

m,Mδ

P

m,M

R,ε

m,Mδ

m,M

k

R,ε

R,ε

(12)

0 50 100 150

020406080100

x

y

### Figure 5:

Simulations of the Directed Spanning Forest with Boolean holes. The paths with direction ex

and coming from vertices with abscissa 0 ≤x≤5 and their ancestors are represented in bold red lines.

We see on (a) that the Boolean holes mostly deviate the paths of the DSF, as in abscissa 50 ordinate 55.

However, when a path is trapped, as in abscissa 100 ordinate 70, it has to cross the hole.

R,ε

k

m,Mδ

m,Mδ

R,ε

R,ε

m,M

m,Mδ

R,ε

R,ε

in

out

Rm,M

R2

Rm,M

in

out

R,ε

R2

R,ε

X

Y

Z

m,Mδ

Rm,M

R,ε

m,Mδ

in

out

in

out

m,Mδ

out

out

R,ε

in

R,ε

in

m,Mδ

in

out

in

out

R,ε

in

out

R,ε

in

out

m,Mδ

out

out

R,ε

in

R,ε

in

P

m,M

P

m,Mδ

R,ε

R,ε

P

m,Mδ

in

out

in

out

R,ε

in

out

R,ε

in

out

P

m,Mδ

out

out

R,ε

in

R,ε

in

P

m,Mδ

out

out

P

R,ε

in

P

R,ε

in

(13)

P

R,ε

in

P

m,Mδ

out

out

P

m,Mδ

in

out

in

out

P

m,Mδ

P

R,ε

in

P

R,ε

### ≥ e

−λπ((R+r)2−(R−r−ε)2)

x

x

x

Theorem 8.

R2

I

R2

I

P

{0}×R

X

R2

eI,J

X

eI,J

X

I

eI,J

E

{0}×[0,L]

E

e{0}×[0,L],{x}×R

e{0}×[0,L],{x}×R

x>0

E

{0}×[0,L]

E

{0}×[0,1]

(14)

{0}×[0,1]

R

R

e{0}×[0,L],{x}×R

{0}×[0,L]

E

e{0}×[0,L],{x}×R

E

{0}×[0,L]

{0}×[0,L]

{x}×[0,L]

Z

{x}×[0,L]

e{0}×R,{x}×[0,L]

X

n∈Z

### R

e{0}×[nL,(n+1)L],{x}×[0,L]

Z

### , R

e{0}×[nL,(n+1)L],{x}×[0,L]

### and R

e{0}×[0,L],{x}×[−nL,−(n−1)L]

E

{0}×[0,L]

E

{x}×[0,L]

E X

n∈Z

### R

e{0}×[0,L],{x}×[−nL,−(n−1)L]

E

e{0}×[0,L],{x}×R

Z2

E

Z2

Z2

E

1

2

x

Z2

E

Z2

O

Z2

Z2

O

Z2E

### In the same way, we wonder if the DSF constructed on the PPP N admits a dual forest with the same distribution, which will allow to derive Theorem 8 from applying Theorem 1 to this dual forest.

Acknowledgements:

(15)

### References

[1] K.S. Alexander. Percolation and minimal spanning forest in infinite graphs. The Annals of Probability, 23(1):87–104, 1995.

[2] S. Athreya, R. Roy, and A. Sarkar. Random directed trees and forest - drainage networks with dependence. Electronic journal of Probability, 13:2160–2189, 2008. Paper no.71.

[3] F. Bacelli and C. Bordenave. The radial spanning tree of a Poisson point process. Annals of Applied Probability, 17(1):305–359, 2007.

[4] N. Bonichon and J.-F. Marckert. Asymptotic of geometrical navigation on a random set of points of the plane. 2010. Submitted.

[5] R.M. Burton and M.S. Keane. Density and uniqueness in percolation. Comm. Math. Phys., 121:501–

505, 1989.

[6] P. A. Ferrari, C. Landim, and H. Thorisson. Poisson trees, succession lines and coalescing random walks. Annales de l’Institut Henri Poincaré, 40:141–152, 2004. Probabilités et Statistiques.

[7] P. A. Ferrari and L. P. R. Pimentel. Competition interfaces and second class particles. Ann. Probab., 33(4):1235–1254, 2005.

[8] L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar. The brownian web: characterization and convergence. Ann. Probab., 32(4):2857–2883, 2004.

[9] S. Gangopadhyay, R. Roy, and A. Sarkar. Random oriented trees: a model of drainage networks.

Ann. App. Probab., 14(3):1242–1266, 2004.

[10] C. D. Howard and C. M. Newman. Euclidean models of first-passage percolation. Probab. Theory Related Fields, 108(2):153–170, 1997.

[11] C. Licea and C. M. Newman. Geodesics in two-dimensional first-passage percolation. Ann. Probab., 24(1):399–410, 1996.

[12] R. Meester and R. Roy. Continuum percolation, volume 119 of Cambridge Tracts in Mathematics.

Cambridge University Press, Cambridge, 1996.

[13] I. Molchanov. Statistics of the Boolean model for practitioners and mathematicians. Chichester, wiley edition, 1997.

[14] R. Schneider and W. Weil. Stochatic and integral geometry. Probability and its Applications. New York, springer-verlag edition, 2008.

Références

Documents relatifs

Computational results based on randomly generated graphs show that the number of k -branch vertices included in the spanning tree increases with the size of the vertex set V,

This pruning procedure allowed to construct a tree-valued Markov process [2] (see also [10] for an analogous construction in a discrete setting) and to study the record process

2 - Modélisation cinétique des oléfines en mélange avec les composés soufrés Le même modèle cinétique qui a permis d’expliquer la transformation des composés soufrés

Son étude théorique impose de reprendre à la base la théorie de l’absorption et de l’émission des photons par un système atomique et le problème même de la

, u k according to these renewal steps β ℓ , ℓ ≥ 1, in order to obtain the searched decay of the tail distribution for the coalescence time of two paths (Theorem 5.1 of Section 5)

Moreover, when used for xed classes G of graphs having optimum (APSPs) compact routing schemes, the construction of an MDST of G ∈ G may be performed by using interval, linear

Moreover, the fact that the paths between two adjacent vertices share a common edge implies that these vertices are inner vertices in different trees (Proposition 1.2.iii)).

Moreover, the fact that the paths between two adjacent vertices share a common edge implies that these vertices are inner vertices in dierent trees (Proposition 1.1.iii)).