Critical density for Activated Random Walk Lorenzo Taggi Max Planck - PowerPoint PPT Presentation

Critical density for Activated Random Walk Critical density for Activated Random Walk Lorenzo Taggi Max Planck Institute for Mathematics in the Sciences Leipzig, Germany June 24, 2014 1

Critical density for Activated Random Walk Outline 1 Definition 2 On monotonicity and the critical density 3 The Diaconis-Fulton Graphical Representation 2

Critical density for Activated Random Walk Definition: Activated Random Walk Two types of particles, A and S, A particles : continuous time random walk in Z d , with jumps rate 1 , distribution of jumps P ( · ) . S particles : at rest. 3

Critical density for Activated Random Walk Definition: Activated Random Walk Two types of particles, A and S, A particles : continuous time random walk in Z d , with jumps rate 1 , distribution of jumps P ( · ) . S particles : at rest. Interaction: A + S − → 2 A (instantaneously) 3

Critical density for Activated Random Walk Definition: Activated Random Walk Two types of particles, A and S, A particles : continuous time random walk in Z d , with jumps rate 1 , distribution of jumps P ( · ) . S particles : at rest. Interaction: A + S − → 2 A (instantaneously) A − → S, rate λ . 3

Critical density for Activated Random Walk Definition: Activated Random Walk Two types of particles, A and S, A particles : continuous time random walk in Z d , with jumps rate 1 , distribution of jumps P ( · ) . S particles : at rest. Interaction: A + S − → 2 A (instantaneously) A − → S, rate λ . Remark: “2A − → A + S − → 2 A” is not observed. 3

Critical density for Activated Random Walk Definition: Activated Random Walk Two types of particles, A and S, A particles : continuous time random walk in Z d , with jumps rate 1 , distribution of jumps P ( · ) . S particles : at rest. Interaction: A + S − → 2 A (instantaneously) A − → S, rate λ . Remark: “2A − → A + S − → 2 A” is not observed. Initial configuration η ∈ Σ = N Z d . ( η ( x )) x ∈ Z d i.i.d. random variables with E [ η ( x )] = µ < ∞ . 3

Critical density for Activated Random Walk Phase Transition 1 Local Fixation : a.s. for any finite V ⊂ Z d ∃ t V such that there is no activity in V for all t > t V . 2 Activity : there is no local fixation. 4

Critical density for Activated Random Walk About monotonicity Time t 1 1 5

Critical density for Activated Random Walk About monotonicity t' 1 t 2 Time 2 1 6

Critical density for Activated Random Walk About monotonicity t 2 t'' 1 Time t 3 2 1 3 7

Critical density for Activated Random Walk Critical density in d = 1 Definition : µ c = sup { µ ∈ [0 , ∞ ] s.t. ARW starting from ν ( µ ) fixates locally } . Theorem [Rolla - Sidoravicius (2009)] Initial configuration: i.i.d. Poisson random variables with expectation µ . Jumps on nearest neighbours. Then, a) ∃ ! µ c ∈ [0 , ∞ ] λ b) If d = 1 , then µ c ∈ [ 1+ λ , 1] . 8

Critical density for Activated Random Walk Critical density in d = 1 Definition : µ c = sup { µ ∈ [0 , ∞ ] s.t. ARW starting from ν ( µ ) fixates locally } . Theorem [Rolla - Sidoravicius (2009)] Initial configuration: i.i.d. Poisson random variables with expectation µ . Jumps on nearest neighbours. Then, a) ∃ ! µ c ∈ [0 , ∞ ] λ b) If d = 1 , then µ c ∈ [ 1+ λ , 1] . Question : is µ c < 1 ? (Dickmann, Rolla, Sidoravicius - 2010) 8

Critical density for Activated Random Walk Theorem [Taggi (2014)] d = 1 . Jumps distribution P (1) = p , P ( − 1) = 1 − p , p ∈ [0 , 1] . Initial configuration: i.i.d. random variables with expectation µ . 1 Let δ ( p ) = | 2 p − 1 | . Then µ c ≤ 1+ λ +1 . δ ( p ) 9

Critical density for Activated Random Walk Theorem [Taggi (2014)] d = 1 . Jumps distribution P (1) = p , P ( − 1) = 1 − p , p ∈ [0 , 1] . Initial configuration: i.i.d. random variables with expectation µ . 1 Let δ ( p ) = | 2 p − 1 | . Then µ c ≤ 1+ λ +1 . δ ( p ) Theorem [Cabezas - Rolla- Sidoravicius (2013)] d = 1 . Jumps distribution P (1) = 1 . Initial configuration: i.i.d. random variables with expectation µ . λ Then µ c = 1+ λ and there is no fixation at µ = µ c . 9

Critical density for Activated Random Walk Critical density in d > 1 Theorem [Shellef (2010), Amir - Gurel Gurevich (2010)] Any d , any λ . Any bounded jumps distribution. Initial configuration: i.i.d random variables with expectation µ . Then µ c ≤ 1 . 10

Critical density for Activated Random Walk Critical density in d > 1 Theorem [Shellef (2010), Amir - Gurel Gurevich (2010)] Any d , any λ . Any bounded jumps distribution. Initial configuration: i.i.d random variables with expectation µ . Then µ c ≤ 1 . Theorem [Taggi 2014] d ≥ 2 Biased ARW Initial configuration: i.i.d. random variables, η ( x ) = 1 with probability µ and η (0) = 0 with probability 1 − µ . 1 There exists K ( P ( · )) > 0 such that µ c ≤ 1+ λ +1 . K 10

Critical density for Activated Random Walk The case of λ → ∞ Theorem [Cabezas - Rolla- Sidoravicius (2013); Shellef (2010), Amir - Gurel Gurevich (2010)] λ = ∞ , any dimension. Any jumps distribution. Initial configuration: i.i.d. Poisson random variables with expectation µ . Then if µ c = 1 and there is no fixation at µ = 1 . 11

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . x -L L 12

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . x -L L 13

Critical density for Activated Random Walk Diaconis-Fulton representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . s x -L L 14

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . s x -L L 15

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . s x -L L 16

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Jumps distribution of ARW: P (1) = p , P ( − 1) = 1 − p . Stabilization of the set [ − L, L ] . s s s x -L 17

Critical density for Activated Random Walk Diaconis-Fulton graphical representation s s s s s s s s s s s s s s s s s s x -L 18

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Definition : let m η,V ( x ) be the number of instructions that must be used at x ∈ Z d in order to stabilize the initial configuration η ∈ N Z d in the (finite) set V ⊂ Z d . s s s s s s s s s s s s s s s s s s x -L 19

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Space ( N Z , I ) I = { τ j x | x ∈ Z , j ∈ N } Probability measure P ν : p P ν ( “ ← ” ) = 1 − p P ν ( τ j P ν ( τ j λ x = “ → ” ) = 1+ λ , 1+ λ , x = “s” ) = 1+ λ s s s s s s s s s s s s s s s s s s x -L 20

Critical density for Activated Random Walk Diaconis-Fulton graphical representation Lemma [Rolla - Sidoravicius (2009)] Let ν be a translation-invariant, ergodic distribution with finite density ν ( η (0)) . Then, P ν ( the system locally fixates ) = P ν ( lim V ↑ Z d m η,V (0) < ∞ ) ∈ { 0 , 1 } . Proposition (Monotonicity) Consider a realization of the instructions, consider η ≺ η ′ , (finite) V ⊂ V ′ ⊂ Z d . Then ∀ x ∈ Z d , m η,V ( x ) ≤ m η ′ ,V ′ ( x ) . 21

Critical density for Activated Random Walk Thank you for your attention! 21