A hierarchical renormalization model on trees Yueyun Hu (Paris 13) - - PowerPoint PPT Presentation

Motivation Results : rate of convergence Proofs

A hierarchical renormalization model on trees

Yueyun Hu (Paris 13) The 3rd conference Probability and Analysis 15–19 May 2017, Be ¸dlewo, Poland

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Overview

Model (Derrida and Retaux) : Let X0 be random variable taking values in {0, 1, 2, ...}. Define Xn+1

law

= (X (1)

n

+ X (2)

n

− 1)+, ∀n ≥ 0, with two independent copies X (1)

n , X (2) n

• f Xn.

Question : What can we say about the asymptotic behaviors of Xn as n → ∞ ?

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Overview

Model (Derrida and Retaux) : Let X0 be random variable taking values in {0, 1, 2, ...}. Define Xn+1

law

= (X (1)

n

+ X (2)

n

− 1)+, ∀n ≥ 0, with two independent copies X (1)

n , X (2) n

• f Xn.

Question : What can we say about the asymptotic behaviors of Xn as n → ∞ ?

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Outline

1 Motivation

Pinning model on a hierarchical structure Derrida and Retaux’ model

2 Results 3 Proofs

Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

The pining model on Z

(Sn)0≤n≤N, S0 = SN = 0.

WALL WALL WALL

ω0 ω1 ω2 ωi ωN (ωi)0≤i≤N i.i.d., independent of (Sn);

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

The pining model on Z

The measure of polymer of length N : PN,ω(dS) := 1 ZN exp

N

• i=1

ωi1{Si=0}

• ,

ZN is called the partition function [ωi > 0 (i attractive) ; ωi < 0 (i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007).

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

The pining model on Z

The measure of polymer of length N : PN,ω(dS) := 1 ZN exp

N

• i=1

ωi1{Si=0}

• ,

ZN is called the partition function [ωi > 0 (i attractive) ; ωi < 0 (i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007).

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

The pining model on Z

The measure of polymer of length N : PN,ω(dS) := 1 ZN exp

N

• i=1

ωi1{Si=0}

• ,

ZN is called the partition function [ωi > 0 (i attractive) ; ωi < 0 (i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007).

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Pinning model on a hierarchical lattice

Derrida, Hakim and Vannimenus (1992)

1 At level 0, there is a unique bond. 2 Fix an integer B ≥ 2 (for e.g. B = 3) 3 Rule : Each bond gives B branches consisting of 2 bonds each. Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Case B = 3

Level n = 0 Level n = 1 WALL WALL WALL

d0 d1 d0 d1 d2 ω1 ω2

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Case B = 3

level n = 2 d0 d1 d2 d3 d2n ω1 ω2 ω3 ω2n

Wall Wall Wall Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Pinning model on a hierarchical lattice

At level n, each (direct) trajectory (Si)1≤i≤2n (from d0 to d2n) contains 2n bonds. Choose the uniform measure PB,n on all possible trajectories [simple random walk (Si)1≤i≤2n on the hierarchical lattice]. Let (ωi)1≤i≤2n be i.i.d. and independent of (Sn). The partition function Zn := EB,n exp

2n

• i=1

ωi1{Si−1=di−1,Si=di}

• ,

where the expectation is only taken with respect to (Sn).

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Pinning model on a hierarchical lattice

Let Nn := number of trajectories γ from d0 to d2n and Rn :=

• γ:γ0=d0,γ2n=d2n

exp

2n

• i=1

ωi1{γi−1=di−1,γi=di}

• .

Then Zn = Rn

Nn .

Easy to see that Nn+1 = B N2

n,

Rn+1 = R(1)

n R(2) n

+ (B − 1)N2

n,

with two independent copies R(1)

n , R(2) n

• f Rn.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Pinning model on a hierarchical lattice

Then Zn+1 = Rn+1 Nn+1 = Z (1)

n Z (2) n

+ B − 1 B , with two independent copies Z (1)

n , Z (2) n

• f Zn.

Stability : (B, Z) → (B′, Z ′) with B′ :=

B B−1, Z ′ := Z B−1.

See Monthus and Garet (2008), Derrida, Giacomin, Lacoin and Toninelli (2009), Lacoin and Toninelli (2009), Giacomin, Lacoin and Toninelli (2010, 2011), Berger and Toninelli (2013) for the studies of this model [disorder relevance, critical line...]

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Pinning model on a hierarchical lattice

Let Xn := log Zn. Then Xn+1 = log Zn+1 = log e(X (1)

n +X (2) n ) + B − 1

B = X (1)

n

+ X (2)

n

+ log 1 + (B − 1)e−(X (1)

n +X (2) n )

B . Let 1 < B < 2 and define a := − log(B − 1) > 0. If Xn ≥ −a, a.s., then Xn+1 ≥ −a a.s.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

Fix a > 0. For any n ≥ 0, Xn+1

law

= max(X (1)

n

+ X (2)

n , −a),

with two independent copies X (1)

n , X (2) n

• f Xn.

Replacing Xn by Xn + a and taking a = 1, the recursive equation becomes Xn+1

law

= (X (1)

n

+ X (2)

n

− 1)+, ∀n ≥ 0, with two independent copies X (1)

n , X (2) n

• f Xn.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

level 0 level 1 level 2 level n

X(1) X(2) X(3) X(4) X(5) X(6) a b X(1)

1

X(2)

1

X(3)

1

(a + b − 1)+ X(1)

n−1

X(2)

n−1

Xn Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

1 Free energy : F∞ := limn→∞

E(Xn) 2n

∈ [0, ∞) exists. Proof : As Xn

law

= (X (1)

n−1 + X (2) n−1 − 1)+,

2E(Xn−1) ≥ E(Xn) ≥ 2E(Xn−1) − 1, implying that F∞ := lim

n→∞ ↓ E(Xn)

2n = lim

n→∞ ↑ E(Xn) − 1

2n .

2 "Percolation on tree" : Let

X0

law

= (1 − p)δ{0} + pδ{Y }, with 0 ≤ p ≤ 1 and Y > 0 a positive random variable. Define pc := sup{0 ≤ p ≤ 1 : F∞(p) = 0}.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

1 Free energy : F∞ := limn→∞

E(Xn) 2n

∈ [0, ∞) exists. Proof : As Xn

law

= (X (1)

n−1 + X (2) n−1 − 1)+,

2E(Xn−1) ≥ E(Xn) ≥ 2E(Xn−1) − 1, implying that F∞ := lim

n→∞ ↓ E(Xn)

2n = lim

n→∞ ↑ E(Xn) − 1

2n .

2 "Percolation on tree" : Let

X0

law

= (1 − p)δ{0} + pδ{Y }, with 0 ≤ p ≤ 1 and Y > 0 a positive random variable. Define pc := sup{0 ≤ p ≤ 1 : F∞(p) = 0}.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

1 Free energy : F∞ := limn→∞

E(Xn) 2n

∈ [0, ∞) exists. Proof : As Xn

law

= (X (1)

n−1 + X (2) n−1 − 1)+,

2E(Xn−1) ≥ E(Xn) ≥ 2E(Xn−1) − 1, implying that F∞ := lim

n→∞ ↓ E(Xn)

2n = lim

n→∞ ↑ E(Xn) − 1

2n .

2 "Percolation on tree" : Let

X0

law

= (1 − p)δ{0} + pδ{Y }, with 0 ≤ p ≤ 1 and Y > 0 a positive random variable. Define pc := sup{0 ≤ p ≤ 1 : F∞(p) = 0}.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

Question 1 : Value of pc. Example : Y ≡ 2 ; pc = 1/2? Question 2 : Behavior of F∞. If pc < 1, when p ↓ pc, what is the behavior of F∞(p)?

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

Question 1 : Value of pc. Example : Y ≡ 2 ; pc = 1/2? Question 2 : Behavior of F∞. If pc < 1, when p ↓ pc, what is the behavior of F∞(p)?

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Value of pc

Theorem (Collet, Eckman, Glaser and Martin 1984) Suppose that Y ∈ {1, 2, ...}. Then pc = 1 1 + E((Y − 1)2Y ). As example, if Y ≡ 2, then pc = 1

5.

Open question Find pc for a general r.v. Y ∈ R+ ; or even for Y ∈ 1

2N...

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Value of pc

Theorem (Collet, Eckman, Glaser and Martin 1984) Suppose that Y ∈ {1, 2, ...}. Then pc = 1 1 + E((Y − 1)2Y ). As example, if Y ≡ 2, then pc = 1

5.

Open question Find pc for a general r.v. Y ∈ R+ ; or even for Y ∈ 1

2N...

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

Corollary For any general r.v. Y ∈ R+, pc > 0 ⇐ ⇒ E(Y 2Y ) < ∞. Open question Find a probabilistic proof on the above L log L-condition.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida and Retaux (2014)’s model

Corollary For any general r.v. Y ∈ R+, pc > 0 ⇐ ⇒ E(Y 2Y ) < ∞. Open question Find a probabilistic proof on the above L log L-condition.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida’s conjecture on the free energy

Derrida’s conjecture : If pc > 0, then F∞(p) = exp

• − K + o(1)

(p − pc)1/2

• ,

p ↓ pc, for some explicit constant K > 0. Why do we need pc > 0 ? Otherwise, the above conjecture is wrong.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Derrida’s conjecture on the free energy

Derrida’s conjecture : If pc > 0, then F∞(p) = exp

• − K + o(1)

(p − pc)1/2

• ,

p ↓ pc, for some explicit constant K > 0. Why do we need pc > 0 ? Otherwise, the above conjecture is wrong.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Model

Model (Galton-Watson case) Let ν be an integer-valued random variable such that m := E(ν) ∈ (1, ∞). Consider the recursive equation Xn+1

law

= (

ν

• i=1

X (i)

n

− 1)+, where X (1)

n , X (2) n , ..., are i.i.d. copies of Xn, and independent of ν.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Suppose X0

law

= (1 − p)δ0 + pδY , with Y > 0 a.s. (not necessarily integer-valued). Let F∞(p) := lim

n→∞

1 mn E(Xn) ∈ [0, ∞) and define pc := sup{0 ≤ p ≤ 1 : F∞(p) = 0}. Open question What is the value of pc in the Galton-Watson case ? Only known when ν equals some integer a.s. [Collet et al. (1984)].

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Suppose X0

law

= (1 − p)δ0 + pδY , with Y > 0 a.s. (not necessarily integer-valued). Let F∞(p) := lim

n→∞

1 mn E(Xn) ∈ [0, ∞) and define pc := sup{0 ≤ p ≤ 1 : F∞(p) = 0}. Open question What is the value of pc in the Galton-Watson case ? Only known when ν equals some integer a.s. [Collet et al. (1984)].

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Proposition In the Galton-Watson case, pc = 0 if and only if E(YmY ) = ∞. Proof : “If part” is a consequence of Collet et al.’s theorem ; “Only if” part : an argument by induction to control the generating function of Xn.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Pinning model on a hierarchical structure Derrida and Retaux’ model

Proposition In the Galton-Watson case, pc = 0 if and only if E(YmY ) = ∞. Proof : “If part” is a consequence of Collet et al.’s theorem ; “Only if” part : an argument by induction to control the generating function of Xn.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Assumptions

We work under the assumption E(YmY ) = ∞ (i.e. pc = 0). Assume that either P(Y > x) ≈ xαm−x, x → ∞, for some α ≥ −2,

• r

P(Y > x) ≈ θ−x, x → ∞, for some 1 < θ < m. We study the rate of convergence of F∞(p) as p → 0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Exponential decay

Theorem 1 (H., Shi (2017+)) Assume that P(Y > x) ≈ xαm−x, x → ∞, for some α > −2. Then F∞(p) = exp

• − p−(1+o(1))/(2+α)

, p → 0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs

Polynomial decay

Theorem 2 (H., Shi (2017+)) Assume that P(Y > x) ≈ θ−x, x → ∞, for some 1 < θ < m. Then F∞(p) ≈ p

log m log(m/θ) ,

p → 0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

From Xn+1

law

= (ν

i=1 X (i) n

− 1)+, we get that mE(Xn) − 1 ≤ E(Xn+1) ≤ mE(Xn). Then F∞(p) = lim ↑ m−n(E(Xn) − 1 m − 1) = lim ↓ m−nE(Xn). Hence F∞(p) > 0 if and only if ∃n ≥ 1 such that E(Xn) >

1 m−1.

Let n0 := inf{n ≥ 1 : E(Xn) >

1 m−1}, then

F∞(p) ≍ m−n0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

From Xn+1

law

= (ν

i=1 X (i) n

− 1)+, we get that mE(Xn) − 1 ≤ E(Xn+1) ≤ mE(Xn). Then F∞(p) = lim ↑ m−n(E(Xn) − 1 m − 1) = lim ↓ m−nE(Xn). Hence F∞(p) > 0 if and only if ∃n ≥ 1 such that E(Xn) >

1 m−1.

Let n0 := inf{n ≥ 1 : E(Xn) >

1 m−1}, then

F∞(p) ≍ m−n0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

From Xn+1

law

= (ν

i=1 X (i) n

− 1)+, we get that mE(Xn) − 1 ≤ E(Xn+1) ≤ mE(Xn). Then F∞(p) = lim ↑ m−n(E(Xn) − 1 m − 1) = lim ↓ m−nE(Xn). Hence F∞(p) > 0 if and only if ∃n ≥ 1 such that E(Xn) >

1 m−1.

Let n0 := inf{n ≥ 1 : E(Xn) >

1 m−1}, then

F∞(p) ≍ m−n0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

From Xn+1

law

= (ν

i=1 X (i) n

− 1)+, we get that mE(Xn) − 1 ≤ E(Xn+1) ≤ mE(Xn). Then F∞(p) = lim ↑ m−n(E(Xn) − 1 m − 1) = lim ↓ m−nE(Xn). Hence F∞(p) > 0 if and only if ∃n ≥ 1 such that E(Xn) >

1 m−1.

Let n0 := inf{n ≥ 1 : E(Xn) >

1 m−1}, then

F∞(p) ≍ m−n0.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

Proof of the polynomial decay case : When P(X0 > x) ∼ p θ−x, x → ∞, with 1 < θ < m. Prove that n0 ≈ log 1/p log m/θ, p → 0, where n0 := inf{n ≥ 1 : E(Xn) >

1 m−1}.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

Proof of Theorem 2 : Lower bound of F∞(p), i.e. upper bound of n0

Let Mn := max1≤i≤mn X (i)

0 . Then Xn ≥ (Mn − n)+.

Extreme value theory implies that Mn ≈ 1 log θ

n log m − log 1

p

.

Then E(Xn) ≥ E(Mn) − n ≈ (log m log θ − 1)n − log 1

p

log θ > 1 m − 1, for all n > (

1 m−1 + log 1

p

log θ )/(log m log θ − 1) ≈ log(1/p) log m/θ .

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

Consider the generating functions Gn(s) := E(sXn), h(s) := E(sν). Then Gn+1(s) = h(Gn(s)) s + s − 1 s h(Gn(0)) ≤ h(Gn(s)) s + s − 1 s . This gives, by iteration, an upper bound on Gn(s) − 1. Choose s = s(p) ∼ m gives a lower bound for n0, i.e. an upper bound for F∞(p). The proof of the upper bound in Theorem 1 is similar, except for the case α ∈ (−2, −1) which is a slightly more delicate.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2

Consider the generating functions Gn(s) := E(sXn), h(s) := E(sν). Then Gn+1(s) = h(Gn(s)) s + s − 1 s h(Gn(0)) ≤ h(Gn(s)) s + s − 1 s . This gives, by iteration, an upper bound on Gn(s) − 1. Choose s = s(p) ∼ m gives a lower bound for n0, i.e. an upper bound for F∞(p). The proof of the upper bound in Theorem 1 is similar, except for the case α ∈ (−2, −1) which is a slightly more delicate.

Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

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