DerridaRetaux model: from discrete to continuous time Michel Pain - - PowerPoint PPT Presentation

Derrida–Retaux model: from discrete to continuous time

Michel Pain (LPSM – DMA) joint work with Yueyun Hu and Bastien Mallein Les Probabilités de demain 14 June 2019

Discrete-time Derrida–Retaux model

Defjnition

⊲ Introduced by Collet–Eckmann–Glaser–Martin (1984), motivated by spin glass theory. Re-introduced by Derrida–Retaux (2014) for studying the depinning transition. Defjnition: Start with a nonnegative random variable X0 and, for any n 0, Xn

1

Xn Xn 1 where Xn is an independent copy of Xn.

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Defjnition

⊲ Introduced by Collet–Eckmann–Glaser–Martin (1984), motivated by spin glass theory. ⊲ Re-introduced by Derrida–Retaux (2014) for studying the depinning transition. Defjnition: Start with a nonnegative random variable X0 and, for any n 0, Xn

1

Xn Xn 1 where Xn is an independent copy of Xn.

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Defjnition

⊲ Introduced by Collet–Eckmann–Glaser–Martin (1984), motivated by spin glass theory. ⊲ Re-introduced by Derrida–Retaux (2014) for studying the depinning transition. ⊲ Defjnition: Start with a nonnegative random variable X0 and, for any n ≥ 0, Xn+1 =

• Xn +

Xn − 1

• +

where Xn is an independent copy of Xn.

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Defjnition on a tree

Construction of Xn on a binary tree: n 2 3 1 4 1 3 i.i.d. copies of X0 Xn

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Defjnition on a tree

Construction of Xn on a binary tree: n 2 3 1 4 1 3 1 3 3 2 i.i.d. copies of X0 Xn a b (a + b − 1)+

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Defjnition on a tree

Construction of Xn on a binary tree: n 2 3 1 4 1 3 1 3 3 2 5 1 i.i.d. copies of X0 Xn a b (a + b − 1)+

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Defjnition on a tree

Construction of Xn on a binary tree: n 2 3 1 4 1 3 1 3 3 2 5 1 4 i.i.d. copies of X0 Xn a b (a + b − 1)+

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Defjnition on a tree

Construction of Xn on a binary tree: n 2 3 1 4 1 3 1 3 3 2 5 1 4 3 i.i.d. copies of X0 Xn a b (a + b − 1)+ Xn

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Phase transition

Free energy: F∞ := lim

n→∞

E[Xn] 2n ∈ [0, ∞]. Theorem (Collet–Eckmann–Glaser–Martin 1984): Assume that X0 a.s. (Supercritical) If X02X0 2X0 or 2X0 , then F and Xn 2n

a.s. n

F (Subcritical) If X02X0 2X0 , then F and Xn

probability n 3/12

Phase transition

Free energy: F∞ := lim

n→∞

E[Xn] 2n ∈ [0, ∞]. Theorem (Collet–Eckmann–Glaser–Martin 1984): Assume that X0 ∈ N a.s. (Supercritical) If X02X0 2X0 or 2X0 , then F and Xn 2n

a.s. n

F (Subcritical) If X02X0 2X0 , then F and Xn

probability n 3/12

Phase transition

Free energy: F∞ := lim

n→∞

E[Xn] 2n ∈ [0, ∞]. Theorem (Collet–Eckmann–Glaser–Martin 1984): Assume that X0 ∈ N a.s. ⊲ (Supercritical) If E

• X02X0

> E

• 2X0
• r E
• 2X0

= ∞, then F∞ > 0 and Xn 2n

a.s.

− − − →

n→∞ F∞.

(Subcritical) If X02X0 2X0 , then F and Xn

probability n 3/12

Phase transition

Free energy: F∞ := lim

n→∞

E[Xn] 2n ∈ [0, ∞]. Theorem (Collet–Eckmann–Glaser–Martin 1984): Assume that X0 ∈ N a.s. ⊲ (Supercritical) If E

• X02X0

> E

• 2X0
• r E
• 2X0

= ∞, then F∞ > 0 and Xn 2n

a.s.

− − − →

n→∞ F∞.

⊲ (Subcritical) If E

• X02X0

≤ E

• 2X0

< ∞, then F∞ = 0 and Xn

probability

− − − − − − →

n→∞

0.

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Free energy near criticality

⊲ Let µ be a probability measure on (0, ∞), in the supercritical phase. Consider X0

(d) 1

p p for each p 0 1 . Let F p denote the free energy and pc inf p 0 1 F p 0 . p F p pc 1 1

2

pc

1 5

Conjecture (Derrida–Retaux 2014): If pc 0, F p exp K

• 1

p pc 1 2 . Theorem (Chen–Dagard–Derrida–Hu–Lifshits–Shi 2019+): If is supported by and x32x dx , then as p pc F p exp 1 p pc 1 2

• 1

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Free energy near criticality

⊲ Let µ be a probability measure on (0, ∞), in the supercritical phase. ⊲ Consider X0

(d)

= (1 − p)δ0 + pµ for each p ∈ [0, 1]. Let F p denote the free energy and pc inf p 0 1 F p 0 . p F p pc 1 1

2

pc

1 5

Conjecture (Derrida–Retaux 2014): If pc 0, F p exp K

• 1

p pc 1 2 . Theorem (Chen–Dagard–Derrida–Hu–Lifshits–Shi 2019+): If is supported by and x32x dx , then as p pc F p exp 1 p pc 1 2

• 1

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Free energy near criticality

⊲ Let µ be a probability measure on (0, ∞), in the supercritical phase. ⊲ Consider X0

(d)

= (1 − p)δ0 + pµ for each p ∈ [0, 1]. ⊲ Let F∞(p) denote the free energy and pc := inf{p ∈ [0, 1] : F∞(p) > 0}. p F∞(p) pc 1 1 µ = δ2 pc = 1

5

Conjecture (Derrida–Retaux 2014): If pc 0, F p exp K

• 1

p pc 1 2 . Theorem (Chen–Dagard–Derrida–Hu–Lifshits–Shi 2019+): If is supported by and x32x dx , then as p pc F p exp 1 p pc 1 2

• 1

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Free energy near criticality

⊲ Let µ be a probability measure on (0, ∞), in the supercritical phase. ⊲ Consider X0

(d)

= (1 − p)δ0 + pµ for each p ∈ [0, 1]. ⊲ Let F∞(p) denote the free energy and pc := inf{p ∈ [0, 1] : F∞(p) > 0}. p F∞(p) pc 1 1 µ = δ2 pc = 1

5

Conjecture (Derrida–Retaux 2014): If pc > 0, F∞(p) = exp

• − K + o(1)

(p − pc)1/2

• .

Theorem (Chen–Dagard–Derrida–Hu–Lifshits–Shi 2019+): If is supported by and x32x dx , then as p pc F p exp 1 p pc 1 2

• 1

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Free energy near criticality

⊲ Let µ be a probability measure on (0, ∞), in the supercritical phase. ⊲ Consider X0

(d)

= (1 − p)δ0 + pµ for each p ∈ [0, 1]. ⊲ Let F∞(p) denote the free energy and pc := inf{p ∈ [0, 1] : F∞(p) > 0}. p F∞(p) pc 1 1 µ = δ2 pc = 1

5

Conjecture (Derrida–Retaux 2014): If pc > 0, F∞(p) = exp

• − K + o(1)

(p − pc)1/2

• .

Theorem (Chen–Dagard–Derrida–Hu–Lifshits–Shi 2019+): If µ is supported by N∗ and ∞ x32xµ(dx) < ∞, then as p ↓ pc F∞(p) = exp

• −

1 (p − pc)1/2+o(1)

• .

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Behavior at criticality

⊲ Critical case for X0 ∈ N: E

• X02X0

= E

• 2X0

< ∞. Recall that Xn 0 in probability. Theorem (Chen–Derrida–Hu–Lifshits–Shi 2017): If X3

02X0

, then Xn c n Conjecture (Chen–Derrida–Hu–Lifshits–Shi 2017): If X3

02X0

, then Xn 4 n2 Given Xn 0, what does the subtree bringing mass to the root look like?

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Behavior at criticality

⊲ Critical case for X0 ∈ N: E

• X02X0

= E

• 2X0

< ∞. ⊲ Recall that Xn → 0 in probability. Theorem (Chen–Derrida–Hu–Lifshits–Shi 2017): If X3

02X0

, then Xn c n Conjecture (Chen–Derrida–Hu–Lifshits–Shi 2017): If X3

02X0

, then Xn 4 n2 Given Xn 0, what does the subtree bringing mass to the root look like?

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Behavior at criticality

⊲ Critical case for X0 ∈ N: E

• X02X0

= E

• 2X0

< ∞. ⊲ Recall that Xn → 0 in probability. ⊲ Theorem (Chen–Derrida–Hu–Lifshits–Shi 2017): If E

• X3

02X0

< ∞, then P(Xn > 0) ≤ c n. Conjecture (Chen–Derrida–Hu–Lifshits–Shi 2017): If X3

02X0

, then Xn 4 n2 Given Xn 0, what does the subtree bringing mass to the root look like?

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Behavior at criticality

⊲ Critical case for X0 ∈ N: E

• X02X0

= E

• 2X0

< ∞. ⊲ Recall that Xn → 0 in probability. ⊲ Theorem (Chen–Derrida–Hu–Lifshits–Shi 2017): If E

• X3

02X0

< ∞, then P(Xn > 0) ≤ c n. ⊲ Conjecture (Chen–Derrida–Hu–Lifshits–Shi 2017): If E

• X3

02X0

< ∞, then P(Xn > 0) ∼ 4 n2 . Given Xn 0, what does the subtree bringing mass to the root look like?

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Behavior at criticality

⊲ Critical case for X0 ∈ N: E

• X02X0

= E

• 2X0

< ∞. ⊲ Recall that Xn → 0 in probability. ⊲ Theorem (Chen–Derrida–Hu–Lifshits–Shi 2017): If E

• X3

02X0

< ∞, then P(Xn > 0) ≤ c n. ⊲ Conjecture (Chen–Derrida–Hu–Lifshits–Shi 2017): If E

• X3

02X0

< ∞, then P(Xn > 0) ∼ 4 n2 . ⊲ Given Xn > 0, what does the subtree bringing mass to the root look like?

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Continuous-time Derrida–Retaux model

Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

When two painters meet, they put their remaining paint in common. Xt is the remaining paint at the root. Xt t

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Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: ⊲ Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

When two painters meet, they put their remaining paint in common. Xt is the remaining paint at the root. Xt t

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Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: ⊲ Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). ⊲ Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

When two painters meet, they put their remaining paint in common. Xt is the remaining paint at the root. Xt t

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Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: ⊲ Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). ⊲ Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. ⊲ Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

When two painters meet, they put their remaining paint in common. Xt is the remaining paint at the root. Xt t

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Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: ⊲ Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). ⊲ Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. ⊲ Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

⊲ When two painters meet, they put their remaining paint in common. Xt is the remaining paint at the root. Xt t

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Defjnition

Initial condition: a nonnegative random variable X0. For t > 0, Xt is defjned using a painting procedure: ⊲ Consider a Yule tree of height t (bi- nary tree with i.i.d. exponentially dis- tributed lifetimes). ⊲ Initially: painters start on the leaves with i.i.d. amount of paint chosen ac- cording to the law of X0. ⊲ Then, painters climb down the tree, painting the branches with a quantity 1

• f paint per unit of branch length.

⊲ When two painters meet, they put their remaining paint in common. ⊲ Xt is the remaining paint at the root. Xt t Xt

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An exactly solvable family of solutions

⊲ From now on, consider X0

(d)

= p0δ0(dx) + (1 − p0)λ0e−λ0x dx. Proposition: For any t 0, Xt

(d p t 0 dx

1 p t t e

t x dx,

where p 0 1 and are the unique solutions of the ODE p 1 p p 1 p with p 0 p0 H p t t log t is an invariant of the dynamics.

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An exactly solvable family of solutions

⊲ From now on, consider X0

(d)

= p0δ0(dx) + (1 − p0)λ0e−λ0x dx. ⊲ Proposition: For any t ≥ 0, Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx, where p: R+ → [0, 1] and λ: R+ → R+ are the unique solutions of the ODE

• p′ = (1 − p)(λ − p)

λ′ = −λ(1 − p) with

• p(0) = p0

λ(0) = λ0. H p t t log t is an invariant of the dynamics.

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An exactly solvable family of solutions

⊲ From now on, consider X0

(d)

= p0δ0(dx) + (1 − p0)λ0e−λ0x dx. ⊲ Proposition: For any t ≥ 0, Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx, where p: R+ → [0, 1] and λ: R+ → R+ are the unique solutions of the ODE

• p′ = (1 − p)(λ − p)

λ′ = −λ(1 − p) with

• p(0) = p0

λ(0) = λ0. ⊲ H := p(t) λ(t) + log λ(t) is an invariant of the dynamics.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

p 1 1 e One can make explicit computations: Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

λ p 1 1 e One can make explicit computations: Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical

λ p 1 1 e One can make explicit computations: Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical

λ p 1 1 e

critical

• One can make explicit computations:

Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical supercritical

λ p 1 1 e

critical

• One can make explicit computations:

Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical supercritical

λ p 1 1 e

critical

• One can make explicit computations:

Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical supercritical

λ p 1 1 e

critical

• p → pc

One can make explicit computations: ⊲ Infjnite order transition for the free energy with exponent 1

2.

Precise asymptotic behavior of p t and t in each phase.

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The phase transition

Xt

(d)

= p(t)δ0(dx) + (1 − p(t))λ(t)e−λ(t)x dx We have p(t) = Hλ(t) − λ(t) log λ(t) with H = p0

λ0 + log λ0.

subcritical supercritical

λ p 1 1 e

critical

• p → pc

One can make explicit computations: ⊲ Infjnite order transition for the free energy with exponent 1

2.

⊲ Precise asymptotic behavior of p(t) and λ(t) in each phase.

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Behavior at criticality

Theorem: With a critical initial condition, P(Xt > 0) = 2 t2 + 16 log t 3t3 + o log t t3

• .

Question: Given Xt 0, what does the subtree bringing paint to the root look like? Xt t the red tree amount of paint along the branches

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Behavior at criticality

Theorem: With a critical initial condition, P(Xt > 0) = 2 t2 + 16 log t 3t3 + o log t t3

• .

Question: Given Xt > 0, what does the subtree bringing paint to the root look like? Xt > 0 t the red tree + amount of paint along the branches

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Convergence of the red tree

⊲ Description from root to leaves of the red tree as a time-inhomogeneous branching Markov pro- cess. Renormalizing time and masses by t, it converges to an explicit limit.

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Convergence of the red tree

⊲ Description from root to leaves of the red tree as a time-inhomogeneous branching Markov pro- cess. ⊲ Renormalizing time and masses by t, it converges to an explicit limit.

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Number and total mass of red leaves

Let Nt be the number of leaves in the red tree of height t and Mt their total mass. Theorem: Given Xt 0, Nt t2 Mt t2 converges in law to an explicit limit.

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Number and total mass of red leaves

Let Nt be the number of leaves in the red tree of height t and Mt their total mass. Theorem: Given Xt > 0, Nt t2 , Mt t2

• converges in law to an explicit limit.

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Thank you for your attention!

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