Introduction G β graphs The rolling Ball Method The main result Proof Continuum Percolation in the β skeleton graph Jean-Michel Billiot, Franck Corset and Eric Fontenas 1 LJK, FIGAL Team Grenoble University SSIAB, 9 may 2012, Avignon Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof Outline Introduction 1 G β graphs 2 The rolling Ball Method 3 The main result 4 Proof 5 Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β 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 R 2 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β 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 of 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 Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β 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 Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof Graphs G β = ( V , E , N β ) ( u , v ) ∈ E ⇔ L u , v ( β ) ∩ V = ∅ respectively C u , v ( β ) ∩ V = ∅ . � c 1 = u + β ( α ) ( v − u ) , αβ ( α ) � L u , v ( β ) = D 2 2 � c 2 = v + ( u − v ) β ( α ) , αβ ( α ) � ∩ D 2 2 � � � � c 1 , αβ ( α ) c 2 , αβ ( α ) C u , v ( β ) = D ∪ D 2 2 with δ ( c 1 , u ) = δ ( c 1 , v ) = δ ( c 2 , u ) = δ ( c 2 , v ) = α β ( α ) and β ( α ) ≥ 1. 2 For 0 < β ( α ) ≤ 1 : � α � � α � C u , v ( β ) = D c 1 , ∩ D c 2 , 2 β ( α ) 2 β ( α ) Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
b b b b b b b b b b b b Introduction G β graphs The rolling Ball Method The main result Proof c 1 c 1 α αβ ( α ) 2 β ( α ) 2 α α c 1 u v u v c 2 α u v c 2 c 2 L u , v ( β ) with β ≥ 1 C u , v ( β ) with β < 1 C u , v ( β ) with β > 1 Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β 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 Z 2 . In these models, the states of the edges are not be independent. Definition A bond percolation model is 1-independent if whenever E 1 and E 2 are sets of edges at graph distance at least 1 from each another (i.e., if no edge of E 1 is incident to an edge of E 2 ) then the state of the edges in E 1 is independent of the state of the edges in E 2 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β 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 Z 2 . In these models, the states of the edges are not be independent. Definition A bond percolation model is 1-independent if whenever E 1 and E 2 are sets of edges at graph distance at least 1 from each another (i.e., if no edge of E 1 is incident to an edge of E 2 ) then the state of the edges in E 1 is independent of the state of the edges in E 2 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
b b Introduction G β graphs The rolling Ball Method The main result Proof The Rolling Ball Method S 1 S 2 q v u r D 1 D v D 2 r L q Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof Comparison with Z 2 Write u ∼ v if uv is an edge of the underlying graph Percolation = infinite path : a sequence u 1 , u 2 . . . with u i ∼ u i + 1 for all i . Let E S 1 , S 2 be the event that every vertex u 1 in the central disk C 1 of S 1 is joined to at least one vertex v in the central disk C 2 of S 2 by a G β − path, regardless of the state of the Poisson process outside of S 1 and S 2 . Each vertex ( i , j ) ∈ Z 2 corresponds to a square [ Ri , R ( i + 1 )] × [ Rj , R ( j + 1 )] ∈ R 2 , where R = 2 r + 2 q , and an edge is open between adjacent vertices (corresponding to squares S 1 and S 2 ) if both events E S 1 , S 2 and E S 2 , S 1 hold. 1-independent model on Z 2 since the event E S 1 , S 2 depends only on the Poisson process within the region S 1 and S 2 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof Comparison with Z 2 Any open path in Z 2 corresponds to a sequence of events E S 1 , S 2 , E S 2 , S 3 . . . that occur, where S i is the square associated with a site in Z 2 . Every vertex u 1 of the original Poisson process that lies in the central disk C 1 of S 1 now has an infinite path leading away from it, since one can find points u i in the central disk of S i and paths from u i − 1 to u i 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. Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof A result of a 1-independent bond percolation on Z 2 Theorem (Balister, Bollobas, Walters. Random Structures and Algorithms , 2005) If every edge in a 1-independent bond percolation model on Z 2 is open 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. Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof The main result Let E S 1 , S 2 be the event that for every point v ∈ C 1 ∪ L , there is a u such that : a) v ∼ u ; b) d ( u , v ) ≤ s ; and c) u ∈ D v , where D v is the disk of radius r inside C 1 ∪ L ∪ C 2 with v on its C 1 -side boundary (the dotted disk in Figure 1). If E S 1 , S 2 holds, then every vertex v in C 1 must be joined by a G β − path to a vertex in C 2 , since each vertex in C 1 ∪ L is joined to a vertex whose disk D v is further along in C 1 ∪ L ∪ C 2 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof The main result E S 1 , S 2 = { ϕ ∈ Ω / ∀ v ∈ ϕ C 1 ∪ L , ∃ u ∈ ϕ Dv ∩ D ( v , s ) , ( ϕ − δ v − δ u )( N β ( uv )) = 0 } A 1 = { ϕ ∈ Ω /ϕ ( D 0 ) > 0 } A = E S 1 , S 2 ∩ E S 2 , S 1 ∩ A 1 Theorem We can find s, r and β , function of the length of edges, so that p (¯ A ) ≤ 0 . 1361 . Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof E S 1 , S 2 ∪ ¯ ¯ A 1 ⊂ ¯ A 1 ∪ A 2 ∪ A 3 A 2 = { ϕ ∈ Ω / ∃ v ∈ ϕ C 1 ∪ L , ( ϕ − δ v )( D v ∩ D ( v , s )) = 0 } . A 3 = { ϕ ∈ Ω / ∃ v ∈ ϕ C 1 ∪ L , ∀ u ∈ ϕ Dv ∩ D ( v , s ) , ( ϕ − δ v − δ u )( N β ( uv )) > 0 } . P (¯ A 1 ) = e − π r 2 . Using Campbell’s theorem and Slyvnyak’s theorem : Given A D v = { ϕ ∈ Ω /ϕ ( D v ∩ D ( v , s )) = 0 } and A D 0 = { ϕ ∈ Ω /ϕ ( D O ∩ D ( O , s )) = 0 } , it comes � 1 A 2 ( ϕ ) ≤ 1 [ C 1 ∪ L ] ( v ) 1 A Dv ( ϕ − δ v ) . v ∈ ϕ P ( A 2 ) ≤ | C 1 ∪ L | P ! O ( A D 0 ) = | C 1 ∪ L | P ( A D 0 ) = 2 r ( 2 r + 2 s ) e −| D O ∩ D ( O , s ) | For the last probability, by introducing the following events Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
Introduction G β graphs The rolling Ball Method The main result Proof A v = { ϕ ∈ Ω / ∀ u ∈ ϕ Dv ∩ D ( v , s ) , ( ϕ − δ u )( N β ( uv )) > 0 } A O = { ϕ ∈ Ω / ∀ u ∈ ϕ DO ∩ D ( O , s ) , ( ϕ − δ u )( N β ( uO )) > 0 } A Ou = { ϕ ∈ Ω /ϕ ( N β ( Ou )) > 0 } . � 1 A 3 ( ϕ ) = max v ∈ ϕ 1 [ C 1 ∪ L ] ( v ) 1 A v ( ϕ − δ v ) ≤ 1 [ C 1 ∪ L ] ( v ) 1 A v ( ϕ − δ v ) . v ∈ ϕ P ( A 3 ) ≤ | C 1 ∪ L | P ! O ( A 0 ) = | C 1 ∪ L | P ( A O ) . � 1 A O ( ϕ ) ≤ 1 DO ∩ D ( O , s ) ( u ) 1 A Ou ( ϕ − δ u ) , u ∈ ϕ � � P ! ( 1 − e −| N β ( Ou ) | ) du . P ( A O ) ≤ u ( A Ou ) du = D O ∩ D ( O , s ) D O ∩ D ( O , s ) � ( 1 − e −| N β ( Ou ) | ) du . P ( A 3 ) ≤ | C 1 ∪ L | D O ∩ D ( O , s ) Continuum Percolation in the β skeleton graph SSIAB 2012, Avignon
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