Condensation and symmetry-breaking in the zero-range process with weak disorder Cécile Mailler Prob@LaB – University of Bath joint work with Peter Mörters (Bath) and Daniel Ueltschi (Warwick) June 9th, 2015 Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 1 / 19
Introduction What is this talk about? Condensation = emergence of a macroscopic phenomenon in a stochastic system. Examples: a node of high degree on a random tree/graph a site containing a lot of particles in a particles system In this talk: The zero-range process (ZRP) in random environment. Outline: the homogenenous ZRP - state of the art 1 ZRP in random environment - different interesting condensation 2 phases Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 2 / 19
The homogeneous ZRP The homogeneous ZRP + [Grosskinsky-Schütz-Spohn ’03] Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 3 / 19
The homogeneous ZRP The zero-range process What is the ZRP? Continuous time Markov process: m particles moving on n sites. Function g ∶ ◆ → ❘ encodes the 1 n 2 interactions between the particles. 3 Stationary distribution: g (2) g (4) g (1) n 1 4 P m , n ( q 1 ,..., q n ) = ∏ p q i 1 q 1 +⋯+ q n = m Z m , n i = 1 where p k = ∏ k − 1 1 j = 1 g ( k ) A purely probabilistic description: Take ( Q i ) i ≥ 1 i.i.d. integer-valued random variables: P ( Q i = k ) = p k . The law of ( Q 1 ,..., Q n ∣ ∑ n i = 1 Q i = m ) is P m , n . Aim: understand P m , n when m , n → +∞ . Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 4 / 19
The homogeneous ZRP Condensation A purely probabilistic description: Take ( Q i ) i ≥ 1 i.i.d. integer-valued random variables: P ( Q i = k ) = p k . The law of ( Q 1 ,..., Q n ∣ ∑ n i = 1 Q i = m ) is P m , n . Notations: LLN implies ∑ n i = 1 Q i ∼ ρ ⋆ n where ρ ⋆ = ❊ Q 1 “ natural density ” We assume that m = m n depends on n , and that m / n → ρ “ forced density ” Theorem [Janson (2012)]: condensation Assume that there exists β > 2 such that p k ∼ k − β ( k → +∞ ), and that ρ > ρ ⋆ , then conditionally to S n ∶ = ∑ n i = 1 Q i = m , in probability when n → +∞ , = ( ρ − ρ ⋆ ) n + o ( n ) ( 1 ) ( 2 ) = o ( n ) . Q and Q n n ( 1 ) + further results (fluctuations of Q n ) Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 5 / 19
The in-homogeneous ZRP The ZRP in random environment Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 6 / 19
The in-homogeneous ZRP The ZRP in random environment 1 n 2 m particles moving on n sites 3 Random environment: i.i.d. random g (2) g (4) X 2 variables X i ∈ [ 0 , 1 ] “ fitnesses ” X n g (1) X 3 4 Function g ∶ ◆ → ❘ encodes the interactions between the particles Stationary distribution: n k − 1 P m , n ( q 1 ,..., q n ) = 1 ∏ where p k = ∏ 1 p q i X q i 1 q 1 +⋯+ q n = m i Z m , n g ( k ) i = 1 j = 1 A purely probabilistic description: Take ( X i ) i ≥ 1 i.i.d. random variables on [ 0 , 1 ] . Take ( Q i ) i ≥ 1 independent integer-valued random variables: P ( Q i = k ) ∝ p k X k i . The law of ( Q 1 ,..., Q n ∣ ∑ n i = 1 Q i = m ) is P m , n . Aim: understand P m , n when m , n → +∞ . Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 7 / 19
The in-homogeneous ZRP The ZRP in random environment Precise set-up Let ( X i ) i ≥ 1 i.i.d. random variables law µ on [ 0 , 1 ] . Let ( Q i ) i ≥ 1 independent random variables P X ( Q i = k ) = p k X k where Φ ( z ) = ∑ i p k z k . Φ ( X i ) k ≥ 0 We assume: ( p k ) k ≥ 0 is a probability distribution µ ([ 1 − h , 1 ]) ∼ h → 0 h γ ( γ > 0) “ disorder parameter ” p k ∼ k →+∞ k − β ( β > 1) “ interaction parameter ” Lemma: “Natural density” In probability when n → +∞ , if β + γ > 2, Q i → E ❊ X Q 1 = E [ X 1 Φ ′ ( X 1 ) nS n ∶ = 1 n ] = ∶ ρ ⋆ . ∑ 1 Φ ( X 1 ) n i = 1 Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 8 / 19
The in-homogeneous ZRP Preliminary: a small proof Lemma: “Natural density” In probability when n → +∞ , if β + γ > 2, Q i → E ❊ X Q 1 = E [ X 1 Φ ′ ( X 1 ) n ] = ∶ ρ ⋆ . 1 ∑ Φ ( X 1 ) n i = 1 Proof: if β + γ > 2 then, in P X - probability, 1 needs a NSC for the n n Q i − 1 1 ∑ ∑ weak LLN for ❊ X Q i → 0 . n n independent RV i = 1 i = 1 ( ❊ X Q i ) i ≥ 1 is a sequence of i.i.d. RV: 2 Φ ( X i ) = X i Φ ′ ( X i ) kp k X k ❊ X Q i = ∑ i Φ ( X i ) To conclude: k ≥ 0 Prove that E ❊ X Q 1 < +∞ ν n ∶ = 1 n ∑ ❊ X Q i → E ❊ X Q 1 . LLN gives: n i = 1 Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 9 / 19
The in-homogeneous ZRP Preliminary: a small proof Lemma: “Natural density” In probability when n → +∞ , if β + γ > 2, Q i → E ❊ X Q 1 = E [ X 1 Φ ′ ( X 1 ) n ] = ∶ ρ ⋆ . 1 ∑ Φ ( X 1 ) n i = 1 Proof: Recall that Φ ( z ) = ∑ k ≥ 0 p k z k and p k ∼ k − β : To conclude: if β > 2 then, E ❊ X Q 1 ≤ Φ ′ ( 1 ) < +∞ qed 1 Prove that if β < 2 then φ ′ ( x ) ∼ x → 1 ( 1 − x ) β − 2 E ❊ X Q 1 < +∞ 2 Fix u ∈ ❘ and think u → +∞ : P ( ❊ X Q 1 ≥ u ) = P ( X 1 Φ ′ ( X 1 ) ≥ u ) ≈ P (( 1 − X 1 ) β − 2 ≥ u ) Φ ( X 1 ) β − 2 ) ∼ u − = P ( X 1 ≥ 1 − u 1 γ 2 − β . Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 10 / 19
The in-homogeneous ZRP Preliminary: a small proof Lemma: “Natural density” In probability when n → +∞ , if β + γ > 2, Q i → E ❊ X Q 1 = E [ X 1 Φ ′ ( X 1 ) n ] = ∶ ρ ⋆ . 1 ∑ Φ ( X 1 ) n i = 1 Proof: Recall that Φ ( z ) = ∑ k ≥ 0 p k z k and p k ∼ k − β : To conclude: if β > 2 then, E ❊ X Q 1 ≤ Φ ′ ( 1 ) < +∞ qed 1 Prove that if β < 2 then φ ′ ( x ) ∼ x → 1 ( 1 − x ) β − 2 E ❊ X Q 1 < +∞ 2 Fix u ∈ ❘ and think u → +∞ : P ( ❊ X Q 1 ≥ u ) = P ( X 1 Φ ′ ( X 1 ) ≥ u ) ≈ P (( 1 − X 1 ) β − 2 ≥ u ) Φ ( X 1 ) β − 2 ) ∼ u − = P ( X 1 ≥ 1 − u 1 γ Integrable as soon as 2 − β . 2 − β > 1 ⇔ β + γ > 2 qed γ Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 10 / 19
The in-homogeneous ZRP Main results Main results: condensation Theorem: condensation Assume that β + γ > 2 and that m / n → ρ > ρ ⋆ , then, conditionally to S n = m , in P m , n -probability when n → +∞ , = ( ρ − ρ ⋆ ) n + o ( n ) = o ( n ) . ( 1 ) ( 2 ) Q and Q n n γ if γ > 1 then, the condensate is a.s. located in the largest fitness site: explicit symmetry-breaking = Q I n = X I n . ( 1 ) ( 1 ) 1 Q where X n n intermediate symmetry- if γ ≤ 1 this is no longer true: breaking 01 β where is the condensate ? 2 Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 11 / 19
The in-homogeneous ZRP Main results: weak disorder phase Weak-disorder phase: location of the condensate Let K n be the rank (in term of decreasing fitness) of the site containing the condensate. We already know that K n = 1 a.s. when γ > 1. Location of the condensate when 0 < γ ≤ 1 Assume β + γ > 2 and ρ > ρ ⋆ , then in quenched distribution when n → +∞ , ( n γ − 1 K n ) 1 / γ → Gamma ( γ,ρ − ρ ⋆ ) . Definition: ( Z n ) n ≥ 1 converges in quenched distribution to Z iff: for all u ∈ ❘ , for all ε > 0, when n → +∞ , P (∣ P X ( Z n ≤ u ∣ S n = m ) − P X ( Z ≤ u )∣ > ε ) → 0 . Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 12 / 19
The in-homogeneous ZRP Main results: weak disorder phase Weak-disorder phase: fitness of the condensate Let F n be the fitness of the site containing the condensate. We already know that F n = max n i = 1 X i a.s. when γ > 1. Fitness of the condensate when 0 < γ ≤ 1 Assume β + γ > 2 and ρ > ρ ⋆ , then in quenched distribution when n → +∞ , n ( 1 − F n ) → Gamma ( γ,ρ − ρ ⋆ ) . Definition: ( Z n ) n ≥ 1 converges in quenched distribution to Z iff: for all u ∈ ❘ , for all ε > 0, when n → +∞ , P (∣ P X ( Z n ≤ u ∣ S n = m ) − P X ( Z ≤ u )∣ > ε ) → 0 . Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 13 / 19
The in-homogeneous ZRP Main results: weak disorder phase Weak disorder phase: fluctuations n ρ n ∶ = m n ν n ∶ = 1 ∑ ❊ X Q i → ρ ⋆ n → ρ New notations: and n i = 1 Theorem: fluctuations when γ ≤ 1 Assume that β + γ > 2 and ρ > ρ ⋆ . Then, if β + γ > 3 then, in P -probability, − ( ρ n − ν n ) n ( 1 ) √ n Q n → W , in distribution, where W is a normal random variable. if 2 < β + γ ≤ 3 then, in P -probability, − ( ρ n − ν n ) n ( 1 ) Q n → W β + γ − 1 , 1 n β + γ − 1 in distribution, where W β + γ − 1 is a ( β + γ − 1 ) -stable random variable. Cécile Mailler (Prob@LaB) Inhomogeneous ZRP June 9th, 2015 14 / 19
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