Introduction Supercritical regimes Recursive trees Scale free trees Almost giant clusters for percolation on large trees Jean Bertoin Institut f¨ ur Mathematik Universit¨ at Z¨ urich Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Erd˝ os-R´ enyi random graph model in supercritical regime G n = complete graph with n vertices Bond percolation with parameter p ( n ) ∼ c / n with c > 1 . There is a unique giant component with size ∼ θ ( c ) n The second, third, etc. largest clusters are almost microscopic (size of order ln n ). Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Percolation on trees much simpler than on general graphs or lattices because unique path between two vertices. E.g. percolation on the infinite regular k -tree ∼ branching process with Bin(k,p) reproduction law. Exist infinite clusters if and only if kp > 1. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Here, we consider percolation on a finite tree T n with size n ≫ 1 Percolation parameter p ( n ) depends on n . Two questions: 1) What are the supercritical regimes (existence of a giant component) ? 2) Estimate the sizes of the 2nd, 3rd, ..., largest clusters. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Characterization of supercritical regimes Rooted tree structure T n with vertices { 0 , 1 , . . . , n } . Bernoulli bond percolation on T n with parameter p ( n ). C 0 p ( n ) = size of cluster contains root. C 0 p ( n ) is giant if n − 1 C 0 p ( n ) = ⇒ G �≡ 0. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees ℓ : N → R + with lim n →∞ ℓ ( n ) = ∞ , and c ≥ 0. Consider the regime c p ( n ) = 1 − ℓ ( n ) + o (1 /ℓ ( n )) . ( R c ) V 1 , V 2 , . . . a sequence of i.i.d. uniform vertices. L k , n = length of the tree reduced to V 1 , . . . , V k and the root 0 1 ℓ ( n ) L k , n = ⇒ L k . ( H k ) Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Theorem (i) If ( H k ) holds for all k, then in the regime ( R c ) n − 1 C 0 p ( n ) ⇒ G ( c ) , (1) where E ( G ( c ) k ) = E ( e − cL k ) . (2) (ii) If (1) holds in the regime ( R c ) for all c > 0 with lim c → 0+ G ( c ) = 1 , then ( H k ) is fulfilled for all k and (2) holds. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees Sketch of proof : �� � k � � � ( n + 1) − 1 C 0 p ( n ) L k , n E = E . p ( n ) In the regime ( R c ), p ( n ) ∼ exp( − c /ℓ ( n )) and ( H k ) yields � � �� − c = E ( e − cL k ) . lim exp ℓ ( n ) L k , n n →∞ E Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Supercritical regimes Recursive trees Scale free trees ( H k ) is known to hold for a number of families of (random) trees: - Cayley trees (Aldous) with ℓ ( n ) = √ n and L k ∼ Chi (2 k ). - d -regular trees with ℓ ( n ) = ln n and L k = k / ln d - random recursive trees, binary search trees, ... with ℓ ( n ) = ln n and L k = k . Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Recursive trees A tree on { 0 , 1 , . . . , n } is called recursive if the sequence of vertices along any branch from the root 0 to a leaf is increasing. There are n ! such recursive trees, we pick one of them uniformly at random, denote it by T n . Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Simple algorithm to construct T n : For i = 1 , 2 , . . . , create an edge between i and U ( i ) randomly chosen in { 0 , . . . , i − 1 } , independently of the U ( j ) for j � = i . Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 2 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 3 2 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 3 2 4 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 3 5 2 4 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees 6 3 5 2 4 1 0 Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees ( H k ) holds with ℓ ( n ) = n and then C 0 p ( n ) ∼ e − c n , in the regime p ( n ) = 1 − c ln n + o (1 / ln n ) Denote by C 1 ( n ) ≥ C 2 ( n ) , . . . the sequence of the sizes of the other clusters ranked in the decreasing order. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Almost giant clusters Theorem For every fixed j ≥ 1 , � ln n n C 1 , . . . , ln n � ⇒ ( x 1 , . . . , x j ) n C j where x 1 > x 2 > . . . denotes the sequence of the atoms of a Poisson random measure on (0 , ∞ ) with intensity c e − c x − 2 d x . Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Some remarks The 2nd, 3rd, ... clusters are almost giant (only fail to be giant by a logarithmic factor). 1 / x 1 , 1 / x 2 − 1 / x 1 , . . . , 1 / x j − 1 / x j − 1 are i.i.d. exponential variables with parameter c e − c . In particular 1 / x j has the gamma distribution with parameter ( j , c e − c ). The parameter c only appears through a constant factor in the intensity measure. Maximal intensity for c = 1. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Some remarks The 2nd, 3rd, ... clusters are almost giant (only fail to be giant by a logarithmic factor). 1 / x 1 , 1 / x 2 − 1 / x 1 , . . . , 1 / x j − 1 / x j − 1 are i.i.d. exponential variables with parameter c e − c . In particular 1 / x j has the gamma distribution with parameter ( j , c e − c ). The parameter c only appears through a constant factor in the intensity measure. Maximal intensity for c = 1. Jean Bertoin Almost giant clusters for percolation on large trees
Introduction Recursive construction Supercritical regimes Almost giant clusters Recursive trees Elements of the proof Scale free trees Some remarks The 2nd, 3rd, ... clusters are almost giant (only fail to be giant by a logarithmic factor). 1 / x 1 , 1 / x 2 − 1 / x 1 , . . . , 1 / x j − 1 / x j − 1 are i.i.d. exponential variables with parameter c e − c . In particular 1 / x j has the gamma distribution with parameter ( j , c e − c ). The parameter c only appears through a constant factor in the intensity measure. Maximal intensity for c = 1. Jean Bertoin Almost giant clusters for percolation on large trees
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