Random growth models with possible extinction R egine Marchand, - PowerPoint PPT Presentation

Random growth models with possible extinction R´ egine Marchand, joint work with Olivier Garet and Jean-Baptiste Gou´ er´ e. SPA 2015, Oxford. Institut Elie Cartan, Universit´ e de Lorraine, Nancy, France.

Random growth models Random growth models: cells, crystals, epidemics... Question Description the asymptotic behaviour of the growth model ? Eden’s model [Eden 61] In Z 2 , start from a single occupied site. At each step, choose a site uniformly among empty neighbours of occupied sites, and fill it. Richarson’s model [Richardson 73] Continuous time analogue for Eden’s model. First-passage percolation [Hammersley–Welsh 65] Random perturbation of the graph distance on Z d . Random growth models with possible extinction: to allow sites to swap back and forth between two states: Oriented percolation [Durrett 84] Contact process [Harris 1974] Continuous time analogue for oriented percolation.

Random growth models with possible extinction 1 Oriented percolation and open paths 2 Convergence results for the number of open paths 3 Shape theorems for oriented percolation 4 Back to the number of open paths

Oriented percolation in dimension d + 1 The oriented graph Z d × N . Each vertex has 2 d + 1 children: N ( x − 1 , n + 1) ( x , n + 1) ( x + 1 , n + 1) ( x , n ) (0 , 0) Z d Randomness. Each edge is independently kept with probability p ∈ (0 , 1). P p : corresponding probability measure.

Oriented percolation: pictures Figure: Examples with p = 0 . 7 , 0 . 6 , 0 . 5 , 0 . 4.

Oriented percolation in dimension d + 1 Phase transition: Does there exist infinite open paths? Ω ∞ = { (0 , 0) → ∞} p > − → P p (Ω ∞ ) > 0 ⇔ p c ( d + 1) . Typical questions: 1 Where are typically the extremities of open paths with length n ? ξ n = { x ∈ Z d : (0 , 0) → ( x , n ) } . � Shape Theorem for the set ξ n . 2 At time n , to what extent ξ n depend on the initial configuration ? � Shape Theorem for the coupled zone. 3 How many open paths with length n can we expect ?

Problem: counting open paths in oriented percolation N x , n : number of open paths from (0 , 0) to ( x , n ) � N n = N x , n : x ∈ Z d number of open paths from (0 , 0) to level n . Figure: n = 3 , p = 0 . 6.     0 1 3 1 4 1 0 10 0 1 1 2 1 1 0 6     ( N x , n ) x , n = and ( N n ) n =     0 0 1 0 1 0 0 2     0 0 0 1 0 0 0 1 Question Asymptotic behaviour of N n ?

Counting open paths: mean behaviour and martingale Mean behaviour: E p ( N n ) = (2 d + 1) n p n ; 1 n log E p ( N n ) = log((2 d + 1) p ) . � � N n is a non-negative martingale: [Darling 91] ((2 d + 1) p ) n N n ∃ W ≥ 0 lim ((2 d + 1) p ) n = W P p − a . s . n → + ∞ 1 on the event { W > 0 } : lim n log N n = log((2 d + 1) p ) . n → + ∞ On { W > 0 } , ( N n ) n has the same exponential growth rate as ( E p ( N n )) n . Question When does { W > 0 } occur ? And what if W = 0 ? [Think about the Kesten–Stigum theorem for the Galton-Watson process 66]

Counting open paths: Mean behaviour and martingale 1 On the event { W > 0 } : lim n log N n = log((2 d + 1) p ) . n → + ∞ it is possible that P p (Ω ∞ ) > 0 and P p ( W = 0) = 1: [dimension 1 and 2: Yoshida 08] it is possible that, on the percolation event, lim n → + ∞ 1 n log N n < log((2 d + 1) p ) for some p ’s, lim n → + ∞ 1 n log N n = log((2 d + 1) p ) for some p ’s. [Spread out percolation and dimension large enough: Lacoin 12] Question a.s. asymptotic behaviour of 1 n log N n on the percolation event ? Conditional probability: P p ( . ) = P p ( . | Ω ∞ ) .

Counting open paths: supermultiplicativity property a , b , c ∈ Z d × N such that a → b → c : N a , c ≥ N a , b N b , c N ( − log N a , c ) ≤ ( − log N a , b ) + ( − log N b , c ) . • c subadditivity • stationarity : b N b , c has the same law as N 0 , c − b • a independence: N b , c is independent from N a , b (0 , 0) Z d � 1 � n log N n should converge. n Subadditive ergodic theorems ? [Kingman 68,73; Hammersley 74...] No: log N a , b can be infinite, and thus is not integrable... Convergence is proved for ρ -percolation [Comets–Popov–Vachkovskaia 08] [Kesten–Sidoravicius 10]

Summary Counting open paths with length n in oriented percolation: Mean behaviour: E p ( N n ) = (2 d + 1) n p n . ( − log N a , c ) ≤ ( − log N a , b ) + ( − log N b , c ): � 1 � n log N n should converge. n Because of possible extinction, infinite quantities appear. Question: How do we prove convergence results in this context ?

Random growth models with possible extinction 1 Oriented percolation and open paths 2 Convergence results for the number of open paths 3 Shape theorems for oriented percolation 4 Back to the number of open paths

Global convergence result Almost-sure convergence on Ω ∞ : Behaviour in mean: Theorem (Garet–Gou´ er´ e–Marchand) 1 1 n log E p ( N n ) = log((2 d + 1) p ) . lim n log N n = ˜ α p (0) P p − a . s . n → + ∞ Figure: Representation of 1 n log N n , as a function of n . Values: n max = 300 and p = 0 . 7 , 0 . 43. Black line: log((2 d + 1) p ).

Directional convergence result N nA n Theorem (Garet–Gou´ er´ e–Marchand 15) There exists a concave function ˜ α p such that, for ”every” set A 1 (0 , 0) Z d lim n log N nA , n = sup α p ( x ) ˜ P p − a . s . n → + ∞ x ∈ A � N nA , n = N x , n . x ∈ nA Figure: n = 100 , p = 0 . 6. Color of pixel ( x , k ) proportional to 1 k log N x , k .

Directional convergence result: p slightly supercritical Figure: n = 300 , p = 0 . 45.

Interpretation as a special case of polymers Polymer in random potential ω : Random walk with length n : a path at random among open paths a path at random among paths P n ,ω ( γ ) = 1 γ open in ω 1 . P n ( γ ) = N n ( ω ) (2 d + 1) n N n ( ω ): quenched partition function

Quenched polymer measure P n ,ω ( γ ) = 1 γ open in ω . N n ( ω ) Global convergence → ω -a.s. existence of the quenched free energy: 1 lim n log N n ( ω ) = ˜ α p (0). n → + ∞ Directional convergence → LDP for the quenched polymer measure: 1 1 n log N nA , n ( ω ) lim n log P n ,ω ( γ n ∈ nA ) = lim N n ( ω ) n → + ∞ n → + ∞ = − inf x ∈ A (˜ α p (0) − ˜ α p ( x )) . Open questions N (0 , 0) Is it true that ∀ x \{ 0 R d } α p ( x ) < ˜ ˜ α p (0) ? nA Is ˜ α p strictly concave ? Is ˜ α p continuous in p ? n quenched free energy=annealed free energy ? Z d

Extension to Linear Stochastic Equation (LSE) Counting all paths : Deterministic linear recurrence equations. � ( x , k + 1) N x , k +1 = N y , k y ∼ x ( x − 1 , k ) ( x , k ) ( x + 1 , k ) ”Pascal’s triangle” Counting open paths : Linear stochastic recurrence equations. � a k ( x , k + 1) N x , k +1 = y , x N y , k y ∼ x ( x − 1 , k ) ( x , k ) ( x + 1 , k ) ”Pascal’s triangle” with iid Bernoulli defects. General Linear Stochastic Equations : [Yoshida 08] � a k iid non-negative N x , k +1 = y , x N y , k coefficients y ∼ x Application : Existence of the quenched free energy for polymer in random potential with values in R + ∪ { + ∞} . [Garet-Gou´ er´ e-Marchand 15]

Convergence results for the number of open paths Our global convergence result Theorem 1 lim n log N n = ˜ α p (0) P p − a . s . n → + ∞ relies on the tools we built for proving shape theorems in oriented percolation...

Random growth models with possible extinction 1 Oriented percolation and open paths 2 Convergence results for the number of open paths 3 Shape theorems for oriented percolation 4 Back to the number of open paths

Oriented percolation on Z d × N with p > − → p c ( d + 1) ξ n = { x ∈ Z d : (0 , 0) → ( x , n ) } . Hitting time : t ( x ) = inf { n ≥ 0 : x ∈ ξ n } . Already visited sites : H n = { x ∈ Z d : t ( x ) ≤ n } . ( H n ) n : non-decreasing sequence of random sets. Figure: Percolation cone, dimension 1 + 1. Theorem (Shape theorem) There exists a norm µ p on R d (unit ball: A µ p ), such that � (1 − ε ) A µ p ⊂ H n + [0 , 1] d � ∃ N > 0 ∀ n ≥ N ⊂ (1 + ε ) A µ p = 1 . P p n [Durrett–Griffeath 82, Bezuidenhout–Grimmett 90, Durrett 91, Garet–Marchand 12]

General strategy for proving a shape theorem: Find a quantity s ( x ) characterizing the growth in a direction x with Subadditivity + Stationarity + Integrability. Subadditive ergodic theorem [Kingman 68,73; Hammersley 74; Liggett 85] to obtain directional limits : s ( nx ) E s ( nx ) µ ( x ) = lim = inf . n n n → + ∞ n ≥ 1 x Prove the convergence is uniform in � x � . Examples: [Eden 61] First-passage percolation: [Richardson 73; Cox–Durrett 81, Boivin 90] Brownian motion in random potential: [Sznitmann 94, Mourrat 12] ”Moving particles”: [Alves-Machado-Popov 02, Kesten–Sidoravicius 05,08] Specific difficulty here: extinction is possible. Conditioning on non-extinction can for instance destroy independence.