Continuum Percolation in the skeleton graph Jean-Michel Billiot, - - PowerPoint PPT Presentation

Introduction Gβ graphs The rolling Ball Method The main result Proof

Continuum Percolation in the β skeleton graph

Jean-Michel Billiot, Franck Corset and Eric Fontenas

1LJK, FIGAL Team

Grenoble University

SSIAB, 9 may 2012, Avignon

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Outline

1

Introduction

2

Gβ graphs

3

The rolling Ball Method

4

The main result

5

Proof

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Continuum percolation result in β skeleton graph for Poisson stationary point process with unit intensity in R2.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Some applications

Ferromagnetism (at low temperature) and Ising model Disordered electrical networks (electrical resistance of a mixture

• f two materials)

Cancerology for the study of the growth of tumor when the cancer cells suddently begin to invade healthy tissue. Epidemics and fires in orchards

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Bibliography

Meester and Roy [5] for continuum percolation Häggström and Meester [4] proposed results for continuum percolation problems for the k-nearest neighbor graph under Poisson process Bertin et al. [2] proved the result for the Gabriel graph Bollobás and Riordan [3] critical probability for random Voronoi percolation in the plane is 1/2. Balister and Bollobás [1] gave bounds on k for the k-nearest neighbor graph for percolation

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Graphs Gβ = (V, E, Nβ)

(u, v) ∈ E ⇔ Lu,v(β) ∩ V = ∅ respectively Cu,v(β) ∩ V = ∅. Lu,v(β) = D

• c1 = u + β(α)

2 (v − u), αβ(α) 2

• ∩ D
• c2 = v + (u − v)β(α)

2 , αβ(α) 2

• Cu,v(β) = D
• c1, αβ(α)

2

• ∪ D
• c2, αβ(α)

2

• with δ(c1, u) = δ(c1, v) = δ(c2, u) = δ(c2, v) = α β(α)

2

and β(α) ≥ 1. For 0 < β(α) ≤ 1 : Cu,v(β) = D

• c1,

α 2β(α)

• ∩ D
• c2,

α 2β(α)

• SSIAB 2012, Avignon

Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

b b

c1 c2 u v

b b

α Lu,v(β) with β ≥ 1 u v

b b

c1 c2

b b

α

α 2β(α)

Cu,v(β) with β < 1 u v

b b

c1 c2

b b

α

αβ(α) 2

Cu,v(β) with β > 1

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

1-independent percolation

To prove that continuous percolation occurs, we shall compare the process to various bond percolation models on Z2. In these models, the states of the edges are not be independent. Definition A bond percolation model is 1-independent if whenever E1 and E2 are sets of edges at graph distance at least 1 from each another (i.e., if no edge of E1 is incident to an edge of E2) then the state of the edges in E1 is independent of the state of the edges in E2.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

1-independent percolation

To prove that continuous percolation occurs, we shall compare the process to various bond percolation models on Z2. In these models, the states of the edges are not be independent. Definition A bond percolation model is 1-independent if whenever E1 and E2 are sets of edges at graph distance at least 1 from each another (i.e., if no edge of E1 is incident to an edge of E2) then the state of the edges in E1 is independent of the state of the edges in E2.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

The Rolling Ball Method

q q r r D1 D2 S1 S2

b b

v u

L Dv

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Comparison with Z2

Write u ∼ v if uv is an edge of the underlying graph Percolation = infinite path : a sequence u1, u2 . . . with ui ∼ ui+1 for all i. Let ES1,S2 be the event that every vertex u1 in the central disk C1

• f S1 is joined to at least one vertex v in the central disk C2 of S2

by a Gβ− path, regardless of the state of the Poisson process

• utside of S1 and S2.

Each vertex (i, j) ∈ Z2 corresponds to a square [Ri, R(i + 1)] × [Rj, R(j + 1)] ∈ R2, where R = 2r + 2q, and an edge is open between adjacent vertices (corresponding to squares S1 and S2) if both events ES1,S2 and ES2,S1 hold. 1-independent model on Z2 since the event ES1,S2 depends only

• n the Poisson process within the region S1 and S2.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Comparison with Z2

Any open path in Z2 corresponds to a sequence of events ES1,S2, ES2,S3 . . . that occur, where Si is the square associated with a site in Z2. Every vertex u1 of the original Poisson process that lies in the central disk C1 of S1 now has an infinite path leading away from it, since one can find points ui in the central disk of Si and paths from ui−1 to ui inductively for every i ≥ 1. One can choose r, q and β so that the probability that the intersection of these events is large and then we will apply the theorem of Balister, Bollobas and Walters.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

A result of a 1-independent bond percolation on Z2

Theorem (Balister, Bollobas, Walters. Random Structures and Algorithms, 2005) If every edge in a 1-independent bond percolation model on Z2 is

• pen with probability at least 0.8639, then almost surely there is an

infinite open component. Moreover, for any bounded region, there is almost surely a cycle of open edges surrounding this region.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

The main result

Let ES1,S2 be the event that for every point v ∈ C1 ∪ L, there is a u such that : a) v ∼ u ; b) d(u, v) ≤ s ; and c) u ∈ Dv, where Dv is the disk of radius r inside C1 ∪ L ∪ C2 with v

• n its C1-side boundary (the dotted disk in Figure 1).

If ES1,S2 holds, then every vertex v in C1 must be joined by a Gβ−path to a vertex in C2, since each vertex in C1 ∪ L is joined to a vertex whose disk Dv is further along in C1 ∪ L ∪ C2.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

The main result

ES1,S2 = {ϕ ∈ Ω/∀v ∈ ϕC1∪L, ∃u ∈ ϕDv∩D(v,s), (ϕ−δv−δu)(Nβ(uv)) = 0} A1 = {ϕ ∈ Ω/ϕ(D0) > 0} A = ES1,S2 ∩ ES2,S1 ∩ A1 Theorem We can find s, r and β, function of the length of edges, so that p(¯ A) ≤ 0.1361.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

¯ ES1,S2 ∪ ¯ A1 ⊂ ¯ A1 ∪ A2 ∪ A3 A2 = {ϕ ∈ Ω/∃v ∈ ϕC1∪L, (ϕ − δv)(Dv ∩ D(v, s)) = 0}. A3 = {ϕ ∈ Ω/∃v ∈ ϕC1∪L, ∀u ∈ ϕDv∩D(v,s), (ϕ − δv − δu)(Nβ(uv)) > 0}. P(¯ A1) = e−πr2. Using Campbell’s theorem and Slyvnyak’s theorem : Given ADv = {ϕ ∈ Ω/ϕ(Dv ∩ D(v, s)) = 0} and AD0 = {ϕ ∈ Ω/ϕ(DO ∩ D(O, s)) = 0}, it comes 1A2(ϕ) ≤

• v∈ϕ

1[C1∪L](v)1ADv(ϕ − δv). P(A2) ≤ |C1∪L| P!

O(AD0) = |C1∪L|P(AD0) = 2r(2r+2s)e−|DO∩D(O,s)|

For the last probability, by introducing the following events

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Av = {ϕ ∈ Ω/∀u ∈ ϕDv∩D(v,s), (ϕ − δu)(Nβ(uv)) > 0} AO = {ϕ ∈ Ω/∀u ∈ ϕDO∩D(O,s), (ϕ − δu)(Nβ(uO)) > 0} AOu = {ϕ ∈ Ω/ϕ(Nβ(Ou)) > 0}. 1A3(ϕ) = max

v∈ϕ 1[C1∪L](v)1Av(ϕ − δv) ≤

• v∈ϕ

1[C1∪L](v)1Av(ϕ − δv). P(A3) ≤ |C1 ∪ L| P!

O(A0) = |C1 ∪ L| P(AO).

1AO(ϕ) ≤

• u∈ϕ

1DO∩D(O,s)(u)1AOu(ϕ − δu), P(AO) ≤

• DO∩D(O,s)

P!

u(AOu)du =

• DO∩D(O,s)

(1 − e−|Nβ(Ou)|)du. P(A3) ≤ |C1 ∪ L|

• DO∩D(O,s)

(1 − e−|Nβ(Ou)|)du.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Lemma P(¯ ES1,S2 ∪ ¯ A1) ≤ e−πr2 + 2r(2r + 2q)e−|DO∩D(O,s)| + 4r(2r + 2q) s α arccos α 2r

• (1 − e−|Nβ(α)|) dα.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Remark : we choose the best q so that every neighborhood of two differents points inside C1 ∪ L stay inside the rectangular zone S1 ∪ S2. We are looking for a function β constant on an interval [0, t] and function of α on the interval [t, s] so that |Nβ(α)| = |Nβ(t)| for all α in [t, s]. We have : P(¯ ES1,S2 ∪ ¯ A1) ≤ e−πr2 + 2r(2r + 2q)e−|DO∩D(O,s)| + 4r(2r + 2q) t α arccos α 2r

• (1 − e−|Nβ(α)|) dα

+4r(2r + 2q) s

t

α arccos α 2r

• (1 − e−|Nβ(t)|) dα.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Numerical results

β Nβ r s a ( t = a/100 × s) 1 (Gabriel Graph) Lu,v(1) 1.437 2.625 1.025 2 (RNG Graph) Lu,v(2) 1.491 2.731 0.631 3 Lu,v(3) 1.515 2.824 0.484 2 Cu,v(2) 1.6 2.882 0.176 3 Cu,v(3) 1.7 2.862 0.087 1/2 Cu,v(1/2) 1.4 2.522 2.71 0 < β ≤ 0.001 Cu,v(β) 1.31 2.6 100

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph

Introduction Gβ graphs The rolling Ball Method The main result Proof

Balister P. and Bollobás, B. Percolation in the k-nearest neighbor graph. Manuscript, 2008. Bertin E., Billiot, J.-M. and Drouilhet, R. Continuum Percolation in the Gabriel Graph. Advances in Applied Probability, 34 :689-701, 2002. Bollobás, B. and Riordan 0. Percolation. Cambridge University Press, 2006. Haggstrom, O. and Meester, R. Nearest Neighbor and Hard Sphere Models in Continuum Percolation. Random Structures and Algorithms, 9(3) :295–315, 1996. Meester, R. and Roy, R. Continuum Percolation. Cambridge University Press, 1996.

SSIAB 2012, Avignon Continuum Percolation in the β skeleton graph