Pinning and Wetting Transition for (1+1)-Dimensional Fields with - - PowerPoint PPT Presentation

The Models Free Energy Path Results Proof

Pinning and Wetting Transition for (1+1)-Dimensional Fields with Laplacian Interaction

Francesco Caravenna

caravenna@math.unipd.it

Universit` a degli Studi di Padova

Workshop on Gradient Models and Elasticity University of Warwick, June 11th, 2008

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 1 / 34

The Models Free Energy Path Results Proof

References

◮ [CD1] F. Caravenna and J.-D. Deuschel

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction, Ann. Probab. (to appear)

◮ [CD2] F. Caravenna and J.-D. Deuschel

Scaling limits of (1+1)-dimensional pinning models with Laplacian interaction, preprint (2008).

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 2 / 34

The Models Free Energy Path Results Proof

Outline

• 1. The Models

Introduction Wetting and pinning models

• 2. Free Energy Results

The free energy The phase transition The disordered case

• 3. Path Results

Path results Refined critical scaling limit

• 4. Sketch of the Proof

Integrated random walk Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 3 / 34

The Models Free Energy Path Results Proof

Outline

• 1. The Models

Introduction Wetting and pinning models

• 2. Free Energy Results

The free energy The phase transition The disordered case

• 3. Path Results

Path results Refined critical scaling limit

• 4. Sketch of the Proof

Integrated random walk Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 4 / 34

The Models Free Energy Path Results Proof Introduction

Some motivations

DNA denaturation transition at high temperature

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 5 / 34

The Models Free Energy Path Results Proof Introduction

Some motivations

DNA denaturation transition at high temperature (1+1)-dimensional model: field above an impenetrable wall

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 5 / 34

The Models Free Energy Path Results Proof Introduction

Some motivations

DNA denaturation transition at high temperature (1+1)-dimensional model: field above an impenetrable wall

Energy Entropy

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 5 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The general wetting model

The field ϕ = {ϕi}1≤i≤N in the free case: Pw

0,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Zw

0,N N

• i=1

dϕ+

i ◮ dϕ+ i is the Lebesgue measure on [0, ∞) ◮ HN(ϕ) describes the structure of the chain (to be specified) ◮ Zw 0,N is the normalization constant (partition function)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 6 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The general wetting model

The field ϕ = {ϕi}1≤i≤N in the interacting case: Pw

ε,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Zw

ε,N N

• i=1
• dϕ+

i

+ ε · δ0(dϕi)

• ◮ dϕ+

i is the Lebesgue measure on [0, ∞) ◮ HN(ϕ) describes the structure of the chain (to be specified) ◮ Zw ε,N is the normalization constant (partition function) ◮ δ0(·) is the Dirac mass at zero ◮ ε ≥ 0 is the strength of the pinning interaction

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 6 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The general pinning model

Analogous to the wetting case but without repulsion: dϕ+

i → dϕi

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 7 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The general pinning model

Analogous to the wetting case but without repulsion: dϕ+

i → dϕi

Pp

ε,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Zp

ε,N N

• i=1
• dϕi + ε · δ0(dϕi)
• ◮ dϕi is the Lebesgue measure on R

◮ HN(ϕ) is the same Hamiltonian (to be specified) ◮ Zp ε,N is the normalization constant (partition function) ◮ δ0(·) is the Dirac mass at zero ◮ ε ≥ 0 is the strength of the pinning interaction

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 7 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Pinning VS Wetting

(+ boundary conditions)

N N + 1 −1 {ϕn}n pinning model Pp

ε,N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 8 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Pinning VS Wetting

(+ boundary conditions)

N N + 1 −1 {ϕn}n pinning model Pp

ε,N

N N + 1 −1 {ϕn}n wetting model Pw

ε,N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 8 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Some questions

Once HN(ϕ) is chosen and ε ≥ 0 is fixed:

◮ What are the properties of Pp ε,N and Pw ε,N for large N ?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 9 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Some questions

Once HN(ϕ) is chosen and ε ≥ 0 is fixed:

◮ What are the properties of Pp ε,N and Pw ε,N for large N ? ◮ Is the field localized at ϕ = 0 or delocalized in ϕ = 0?

(How to define localization and delocalization?)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 9 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Some questions

Once HN(ϕ) is chosen and ε ≥ 0 is fixed:

◮ What are the properties of Pp ε,N and Pw ε,N for large N ? ◮ Is the field localized at ϕ = 0 or delocalized in ϕ = 0?

(How to define localization and delocalization?)

◮ Does the answer depend on ε ≥ 0 (phase transition) and/or

• n the choice of HN(ϕ)?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 9 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Some questions

Once HN(ϕ) is chosen and ε ≥ 0 is fixed:

◮ What are the properties of Pp ε,N and Pw ε,N for large N ? ◮ Is the field localized at ϕ = 0 or delocalized in ϕ = 0?

(How to define localization and delocalization?)

◮ Does the answer depend on ε ≥ 0 (phase transition) and/or

• n the choice of HN(ϕ)?

How to choose HN(ϕ) ?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 9 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

The simplest choice is the gradient case: HN(ϕ) :=

N

• i=1

V

• ∇ϕi
• ,

∇ϕi := ϕi − ϕi−1 ,

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 10 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

The simplest choice is the gradient case: HN(ϕ) :=

N

• i=1

V

• ∇ϕi
• ,

∇ϕi := ϕi − ϕi−1 , V (·) : R → R ∪ {+∞} :

• R

e−V (x) dx < +∞ ( = 1 ) . + regularity (see later). Gaussian case: V (x) ∝ x2

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 10 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

The simplest choice is the gradient case: HN(ϕ) :=

N

• i=1

V

• ∇ϕi
• ,

∇ϕi := ϕi − ϕi−1 , V (·) : R → R ∪ {+∞} :

• R

e−V (x) dx < +∞ ( = 1 ) . + regularity (see later). Gaussian case: V (x) ∝ x2

◮ Pp 0,N is (the bridge of) a random walk of step law e−V (x) ◮ Pw 0,N is (the bridge of) a random walk conditioned to stay ≥ 0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 10 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

The simplest choice is the gradient case: HN(ϕ) :=

N

• i=1

V

• ∇ϕi
• ,

∇ϕi := ϕi − ϕi−1 , V (·) : R → R ∪ {+∞} :

• R

e−V (x) dx < +∞ ( = 1 ) . + regularity (see later). Gaussian case: V (x) ∝ x2

◮ Pp 0,N is (the bridge of) a random walk of step law e−V (x) ◮ Pw 0,N is (the bridge of) a random walk conditioned to stay ≥ 0

[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05] [Caravenna, Giacomin, Zambotti EJP 06]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 10 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

We rather consider the Laplacian case: HN(ϕ) :=

N

• i=0

V

• ∆ϕi
• ∆ϕi := ∇ϕi+1 − ∇ϕi = ϕi+1 + ϕi−1 − 2 ϕi

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

We rather consider the Laplacian case: HN(ϕ) :=

N

• i=0

V

• ∆ϕi
• ∆ϕi := ∇ϕi+1 − ∇ϕi = ϕi+1 + ϕi−1 − 2 ϕi

◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

We rather consider the Laplacian case: HN(ϕ) :=

N

• i=0

V

• ∆ϕi
• +

V

• ∇ϕi
• ∆ϕi := ∇ϕi+1 − ∇ϕi = ϕi+1 + ϕi−1 − 2 ϕi

◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations ◮ ∇ and ∆ together?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

The choice of HN(ϕ)

We rather consider the Laplacian case: HN(ϕ) :=

N

• i=0

V

• ∆ϕi
• ∆ϕi := ∇ϕi+1 − ∇ϕi = ϕi+1 + ϕi−1 − 2 ϕi

◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations ◮ ∇ and ∆ together?

Interpretation of the free case ε = 0:

◮ Pp 0,N is (the bridge of) the integral of a random walk ◮ Pw 0,N is further conditioned to stay ≥ 0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Laplacian interaction in (d + 1)-dimension

Fields ϕ : {1, . . . , N}d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Laplacian interaction in (d + 1)-dimension

Fields ϕ : {1, . . . , N}d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case:

[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Laplacian interaction in (d + 1)-dimension

Fields ϕ : {1, . . . , N}d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case:

[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]

Henceforth we study Pp

ε,N and Pw ε,N with Laplacian interaction and

boundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34

The Models Free Energy Path Results Proof Wetting and pinning models

Laplacian interaction in (d + 1)-dimension

Fields ϕ : {1, . . . , N}d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case:

[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]

Henceforth we study Pp

ε,N and Pw ε,N with Laplacian interaction and

boundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0 Assumptions on V :

• R

e−V (x) dx = 1 ,

• R

x e−V (x) dx = 0 ,

• R

x2 e−V (x) dx = 1 + regularity: x → e−V (x) continuous and V (0) < +∞.

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34

The Models Free Energy Path Results Proof

Outline

• 1. The Models

Introduction Wetting and pinning models

• 2. Free Energy Results

The free energy The phase transition The disordered case

• 3. Path Results

Path results Refined critical scaling limit

• 4. Sketch of the Proof

Integrated random walk Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 13 / 34

The Models Free Energy Path Results Proof The free energy

How to define localization and delocalization?

Recall the partition function:

(zero boundary conditions)

Za

ε,N =

• Ωa

N

e−HN(ϕ)

N−1

• i=1
• dϕi + ε δ0(dϕi)
• where a ∈ {p, w} and Ωp

N = RN−1 while Ωw N = [0, ∞)N−1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34

The Models Free Energy Path Results Proof The free energy

How to define localization and delocalization?

Recall the partition function:

(zero boundary conditions)

Za

ε,N =

• Ωa

N

e−HN(ϕ)

N−1

• i=1
• dϕi + ε δ0(dϕi)
• where a ∈ {p, w} and Ωp

N = RN−1 while Ωw N = [0, ∞)N−1

Free Energy

Fa(ε) := lim

N→∞

1 N log Za

ε,N

(super-additivity)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34

The Models Free Energy Path Results Proof The free energy

How to define localization and delocalization?

Recall the partition function:

(zero boundary conditions)

Za

ε,N =

• Ωa

N

e−HN(ϕ)

N−1

• i=1
• dϕi + ε δ0(dϕi)
• where a ∈ {p, w} and Ωp

N = RN−1 while Ωw N = [0, ∞)N−1

Free Energy

Fa(ε) := lim

N→∞

1 N log Za

ε,N

(super-additivity)

Basic observation: Fa(ε) ≥ Fa(0) = 0 for all ε ≥ 0 and a ∈ {p, w} Za

ε,N ≥ Za 0,N ≈ N−c

(c > 0)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

◮ if ε > εa c then ℓN

N → Da(ε) > 0 in Pa

ε,N –probability

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

◮ if ε > εa c then ℓN

N → Da(ε) > 0 in Pa

ε,N –probability ◮ if ε < εa c then ℓN

N → 0 in Pa

ε,N –probability (delocalization)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

◮ if ε > εa c then ℓN

N → Da(ε) > 0 in Pa

ε,N –probability ◮ if ε < εa c then ℓN

N → 0 in Pa

ε,N –probability (delocalization) ◮ if ε = εa c ? Depends on the model:

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

◮ if ε > εa c then ℓN

N → Da(ε) > 0 in Pa

ε,N –probability ◮ if ε < εa c then ℓN

N → 0 in Pa

ε,N –probability (delocalization) ◮ if ε = εa c ? Depends on the model:

◮ if Fa(εa

c + h) = o(h) [> 1st order trans.] ε = εa c is delocalized

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The free energy

Localization and delocalization

Definition

The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0. localized ⇐ ⇒ ε > εa

c := sup{ε ≥ 0 : Fa(ε) = 0} ∈ [0, ∞]

Setting ℓN := #

• i ≤ N : ϕi = 0
• we have:

◮ if ε > εa c then ℓN

N → Da(ε) > 0 in Pa

ε,N –probability ◮ if ε < εa c then ℓN

N → 0 in Pa

ε,N –probability (delocalization) ◮ if ε = εa c ? Depends on the model:

◮ if Fa(εa

c + h) = o(h) [> 1st order trans.] ε = εa c is delocalized

◮ if Fa(εa

c + h) ≥ C h [1st order trans.] ε = εa c may be localized

(phase coexistence, dependence of boundary conditions)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34

The Models Free Energy Path Results Proof The phase transition

The phase transition

Theorem ([CD1])

Both Pp

ε,N and Pw ε,N undergo a non-trivial phase transition:

0 < εp

c < εw c

< ∞ and Fa(ε) is analytic on [0, εa

c) ∪ (εa c, ∞). (variational formula)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34

The Models Free Energy Path Results Proof The phase transition

The phase transition

Theorem ([CD1])

Both Pp

ε,N and Pw ε,N undergo a non-trivial phase transition:

0 < εp

c < εw c

< ∞ and Fa(ε) is analytic on [0, εa

c) ∪ (εa c, ∞). (variational formula) ◮ In the pinning model the transition is exactly of 2nd order:

C1 h log 1

h

≤ Fp(εp

c + h) ≤ o(h)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34

The Models Free Energy Path Results Proof The phase transition

The phase transition

Theorem ([CD1])

Both Pp

ε,N and Pw ε,N undergo a non-trivial phase transition:

0 < εp

c < εw c

< ∞ and Fa(ε) is analytic on [0, εa

c) ∪ (εa c, ∞). (variational formula) ◮ In the pinning model the transition is exactly of 2nd order:

C1 h log 1

h

≤ Fp(εp

c + h) ≤ o(h) ◮ In the wetting model the transition is of 1st order:

Fw(εw

c + h) ∼ C2 h

• ℓN ∼ D N ,

D > 0

• Francesco Caravenna

(1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34

The Models Free Energy Path Results Proof The phase transition

The gradient case

Differences in the gradient case

◮ the transition is non-trivial only in the wetting model:

εp,∇

c

= 0 , 0 < εw,∇

c

< ∞

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34

The Models Free Energy Path Results Proof The phase transition

The gradient case

Differences in the gradient case

◮ the transition is non-trivial only in the wetting model:

εp,∇

c

= 0 , 0 < εw,∇

c

< ∞

◮ the transition is of 2nd order:

Fp εp,∇

c

+ h

• ∼ Cp h2 ,

Fw εw,∇

c

+ h

• ∼ Cw h2

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34

The Models Free Energy Path Results Proof The phase transition

The gradient case

Differences in the gradient case

◮ the transition is non-trivial only in the wetting model:

εp,∇

c

= 0 , 0 < εw,∇

c

< ∞

◮ the transition is of 2nd order:

Fp εp,∇

c

+ h

• ∼ Cp h2 ,

Fw εw,∇

c

+ h

• ∼ Cw h2

∇ ∆ ε εp

c

εp,∇

c

Fp(ε) ∇ ∆ ε εw

c

εw,∇

c

Fw(ε)

skip Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34

The Models Free Energy Path Results Proof The disordered case

A look at the disordered case

Disordered version of our model: (dϕp

i = dϕi and dϕw i = dϕ+ i )

Pa

ε,β,ω,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Za

ε,β,ω,N N

• i=1
• dϕa

i + ε eβωi δ0(dϕi)

• where β ≥ 0 and {ωi}i∈N are IID N(0, 1) (law P indep. Pa).

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34

The Models Free Energy Path Results Proof The disordered case

A look at the disordered case

Disordered version of our model: (dϕp

i = dϕi and dϕw i = dϕ+ i )

Pa

ε,β,ω,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Za

ε,β,ω,N N

• i=1
• dϕa

i + ε eβωi δ0(dϕi)

• where β ≥ 0 and {ωi}i∈N are IID N(0, 1) (law P indep. Pa).

Quenched free energy

Fa ε, β

• :=

lim

N→∞

1 N log Za

ε,β,ω,N ≥ 0

exists P(dω)–a.s. and does not depend on ω (self-averaging)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34

The Models Free Energy Path Results Proof The disordered case

A look at the disordered case

Disordered version of our model: (dϕp

i = dϕi and dϕw i = dϕ+ i )

Pa

ε,β,ω,N

• dϕ1 , . . . , dϕN
• := e−HN(ϕ)

Za

ε,β,ω,N N

• i=1
• dϕa

i + ε eβωi δ0(dϕi)

• where β ≥ 0 and {ωi}i∈N are IID N(0, 1) (law P indep. Pa).

Quenched free energy

Fa ε, β

• :=

lim

N→∞

1 N log Za

ε,β,ω,N ≥ 0

exists P(dω)–a.s. and does not depend on ω (self-averaging) Localization: Fa(ε, β) > 0 ⇐ ⇒ ε > εa

c(β) (critical line)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34

The Models Free Energy Path Results Proof The disordered case

Smoothing effect of disorder

What is the behavior of εa

c(β) for small β ? (εa

c = εa c(0))

What is the regularity of the transition in the disordered case?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34

The Models Free Energy Path Results Proof The disordered case

Smoothing effect of disorder

What is the behavior of εa

c(β) for small β ? (εa

c = εa c(0))

What is the regularity of the transition in the disordered case?

Theorem ([Giacomin and Toninelli, CMP 06])

Both in the ∇ and ∆ case, both for a = p and for a = w: for every β > 0 there exists Cβ > 0 such that Fa εa

c(β) + h

• ≤ Cβ h2

When disorder is present the transition is at least of 2nd order Sharp contrast with homogeneous case

cf. Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34

The Models Free Energy Path Results Proof The disordered case

Smoothing effect of disorder

What is the behavior of εa

c(β) for small β ? (εa

c = εa c(0))

What is the regularity of the transition in the disordered case?

Theorem ([Giacomin and Toninelli, CMP 06])

Both in the ∇ and ∆ case, both for a = p and for a = w: for every β > 0 there exists Cβ > 0 such that Fa εa

c(β) + h

• ≤ Cβ h2

When disorder is present the transition is at least of 2nd order Sharp contrast with homogeneous case

cf.

Very general proof: rare-stretches in ω (Large Deviations)

proof Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34

The Models Free Energy Path Results Proof

Outline

• 1. The Models

Introduction Wetting and pinning models

• 2. Free Energy Results

The free energy The phase transition The disordered case

• 3. Path Results

Path results Refined critical scaling limit

• 4. Sketch of the Proof

Integrated random walk Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 20 / 34

The Models Free Energy Path Results Proof

Some deeper questions

We have established the existence of a phase transition: ℓN =

• (N)

if ε < εa

c

∼ D · N if ε > εa

c

ℓN := #

• i ≤ N : ϕi = 0
• Francesco Caravenna

(1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34

The Models Free Energy Path Results Proof

Some deeper questions

We have established the existence of a phase transition: ℓN =

• (N)

if ε < εa

c

∼ D · N if ε > εa

c

ℓN := #

• i ≤ N : ϕi = 0
• Can we say something more precise on ℓN? And on the maximum

MN := max

1≤i≤N

• ϕi
• Francesco Caravenna

(1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34

The Models Free Energy Path Results Proof

Some deeper questions

We have established the existence of a phase transition: ℓN =

• (N)

if ε < εa

c

∼ D · N if ε > εa

c

ℓN := #

• i ≤ N : ϕi = 0
• Can we say something more precise on ℓN? And on the maximum

MN := max

1≤i≤N

• ϕi
• Yes in the pinning case and under additional assumptions on V (·):

◮ symmetry: V (x) = V (−x) for every x ∈ R ◮ uniform strict convexity: ∃γ > 0 s. t. V (x) − γ x2 2 is convex ◮ regularity: x → e−V (x) is continuous and V (0) < ∞

• R

e−V (x) dx = 1

• R

x2 e−V (x) dx = 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

lim

K→∞ lim inf N→∞ Pp ε,N

1 K N3/2 ≤ MN ≤ K N3/2

• = 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

lim

K→∞ lim inf N→∞ Pp ε,N

1 K N3/2 ≤ MN ≤ K N3/2

• = 1

◮ localized regime ε > εp c : ℓN ∼ DN and MN = O

• (log N)2

:

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

lim

K→∞ lim inf N→∞ Pp ε,N

1 K N3/2 ≤ MN ≤ K N3/2

• = 1

◮ localized regime ε > εp c : ℓN ∼ DN and MN = O

• (log N)2

: lim

K→∞ lim inf N→∞ Pp ε,N

• MN ≤ K (log N)2

= 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

lim

K→∞ lim inf N→∞ Pp ε,N

1 K N3/2 ≤ MN ≤ K N3/2

• = 1

◮ localized regime ε > εp c : ℓN ∼ DN and MN = O

• (log N)2

: lim

K→∞ lim inf N→∞ Pp ε,N

• MN ≤ K (log N)2

= 1

◮ critical regime ε = εp c : ℓN ≈ N log N and MN ≈ N3/2 (log N)c :

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

A closer look at the typical paths [CD2]

◮ delocalized regime ε < εp c : ℓN = O(1) and MN ≈ N3/2 :

lim

K→∞ lim inf N→∞ Pp ε,N

• ϕi = 0 for i ∈ {K, N − K}
• = 1

lim

K→∞ lim inf N→∞ Pp ε,N

1 K N3/2 ≤ MN ≤ K N3/2

• = 1

◮ localized regime ε > εp c : ℓN ∼ DN and MN = O

• (log N)2

: lim

K→∞ lim inf N→∞ Pp ε,N

• MN ≤ K (log N)2

= 1

◮ critical regime ε = εp c : ℓN ≈ N log N and MN ≈ N3/2 (log N)c :

lim

K→∞ lim inf N→∞ Pp εp

c ,N

1 K N3/2 (log N)3/2 ≤ MN ≤ K N3/2 log N

• = 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34

The Models Free Energy Path Results Proof Path results

Scaling Limits

We rescale and interpolate linearly the field: for t ∈ [0, 1]

• ϕN(t) := ϕ⌊Nt⌋

N3/2 + (Nt − ⌊Nt⌋) ϕ⌊Nt⌋+1 − ϕ⌊Nt⌋ N3/2

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34

The Models Free Energy Path Results Proof Path results

Scaling Limits

We rescale and interpolate linearly the field: for t ∈ [0, 1]

• ϕN(t) := ϕ⌊Nt⌋

N3/2 + (Nt − ⌊Nt⌋) ϕ⌊Nt⌋+1 − ϕ⌊Nt⌋ N3/2 Let {Bt}t∈[0,1] standard BM, It := t

0 Bs ds integrated BM

Bridge

• It
• t∈[0,1] := {It}t∈[0,1] conditionally on (B1, I1) = (0, 0)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34

The Models Free Energy Path Results Proof Path results

Scaling Limits

We rescale and interpolate linearly the field: for t ∈ [0, 1]

• ϕN(t) := ϕ⌊Nt⌋

N3/2 + (Nt − ⌊Nt⌋) ϕ⌊Nt⌋+1 − ϕ⌊Nt⌋ N3/2 Let {Bt}t∈[0,1] standard BM, It := t

0 Bs ds integrated BM

Bridge

• It
• t∈[0,1] := {It}t∈[0,1] conditionally on (B1, I1) = (0, 0)

Theorem (Scaling Limits [CD2])

The rescaled field { ϕN(t)}t∈[0,1] under Pp

ε,N converges in

distribution on C([0, 1]) as N → ∞, for every ε ≥ 0. The limit is

◮ If ε < εp c , the law of {

It}t∈[0,1]

◮ If ε = εp c or ε > εp c , the law concentrated on f (t) ≡ 0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

For ε = εp

c the field has very large fluctuations ( ≈ N3/2 (log N)c ). ϕi O

• N

log N

• N

O

• N 3/2

(log N)3/2

• Francesco Caravenna

(1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

For ε = εp

c the field has very large fluctuations ( ≈ N3/2 (log N)c ). ϕi O

• N

log N

• N

O

• N 3/2

(log N)3/2

• Can we extract a non-trivial scaling limit?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

For ε = εp

c the field has very large fluctuations ( ≈ N3/2 (log N)c ). ϕi O

• N

log N

• N

O

• N 3/2

(log N)3/2

• Can we extract a non-trivial scaling limit?

Not in C([0, 1]) or D([0, 1]): the set 1 N

• i ∈ {1, . . . , N} : ϕi = 0
• becomes dense in [0, 1]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

For ε = εp

c the field has very large fluctuations ( ≈ N3/2 (log N)c ). ϕi O

• N

log N

• N

O

• N 3/2

(log N)3/2

• Can we extract a non-trivial scaling limit?

Not in C([0, 1]) or D([0, 1]): the set 1 N

• i ∈ {1, . . . , N} : ϕi = 0
• becomes dense in [0, 1]

Alternative idea: look at the field in a distributional sense

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

Introduce the random measure (finite, signed) on [0, 1] µN

• dt
• := (log N)5/2

N3/2 ϕ⌊Nt⌋ dt = ϕN(t) dt

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

Introduce the random measure (finite, signed) on [0, 1] µN

• dt
• := (log N)5/2

N3/2 ϕ⌊Nt⌋ dt = ϕN(t) dt Let {Lt}t∈[0,1] be the stable symmetric L´ evy process of index 2

5

0 drift 0 Brownian component Π(dx) = c |x|−7/5 dx

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

Introduce the random measure (finite, signed) on [0, 1] µN

• dt
• := (log N)5/2

N3/2 ϕ⌊Nt⌋ dt = ϕN(t) dt Let {Lt}t∈[0,1] be the stable symmetric L´ evy process of index 2

5

0 drift 0 Brownian component Π(dx) = c |x|−7/5 dx The paths of L are a.s. of bounded variation, hence we set dL

• (a, b]
• := Lb − La

random measure on [0, 1]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

Introduce the random measure (finite, signed) on [0, 1] µN

• dt
• := (log N)5/2

N3/2 ϕ⌊Nt⌋ dt = ϕN(t) dt Let {Lt}t∈[0,1] be the stable symmetric L´ evy process of index 2

5

0 drift 0 Brownian component Π(dx) = c |x|−7/5 dx The paths of L are a.s. of bounded variation, hence we set dL

• (a, b]
• := Lb − La

random measure on [0, 1]

Theorem ([CD2])

µN under Pp

εp

c ,N converges in distribution as N → ∞ toward dL . Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34

The Models Free Energy Path Results Proof Refined critical scaling limit

The critical regime

1 1

• ϕN(t)

dL

O

• 1

log N

• O
• log N
• disorder

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 26 / 34

The Models Free Energy Path Results Proof

Outline

• 1. The Models

Introduction Wetting and pinning models

• 2. Free Energy Results

The free energy The phase transition The disordered case

• 3. Path Results

Path results Refined critical scaling limit

• 4. Sketch of the Proof

Integrated random walk Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 27 / 34

The Models Free Energy Path Results Proof Integrated random walk

A random walk viewpoint (ε = 0)

Let {Xi}i∈N be IID random variables with law P(Xi ∈ dx) := e−V (x) dx E(Xi) = 0 Var(Xi) = 1

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34

The Models Free Energy Path Results Proof Integrated random walk

A random walk viewpoint (ε = 0)

Let {Xi}i∈N be IID random variables with law P(Xi ∈ dx) := e−V (x) dx E(Xi) = 0 Var(Xi) = 1 Random walk: Yn := X1 + . . . + Xn Integrated random walk:

entropic repulsion

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34

The Models Free Energy Path Results Proof Integrated random walk

A random walk viewpoint (ε = 0)

Let {Xi}i∈N be IID random variables with law P(Xi ∈ dx) := e−V (x) dx E(Xi) = 0 Var(Xi) = 1 Random walk: Yn := X1 + . . . + Xn Integrated random walk:

entropic repulsion

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

The free case ε = 0

◮ The field {ϕi}1≤i≤N under Pp 0,N is distributed like {Zi}1≤i≤N

conditionally on (YN, ZN) = (0, 0). (ZN/2 ≈ N3/2)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34

The Models Free Energy Path Results Proof Integrated random walk

A random walk viewpoint (ε = 0)

Let {Xi}i∈N be IID random variables with law P(Xi ∈ dx) := e−V (x) dx E(Xi) = 0 Var(Xi) = 1 Random walk: Yn := X1 + . . . + Xn Integrated random walk:

entropic repulsion

Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn

The free case ε = 0

◮ The field {ϕi}1≤i≤N under Pp 0,N is distributed like {Zi}1≤i≤N

conditionally on (YN, ZN) = (0, 0). (ZN/2 ≈ N3/2)

◮ Under Pw 0,N the same, under the further conditioning

{Z1 ≥ 0, . . . , ZN ≥ 0}

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions.

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N Auxiliary chain: Ji := ϕτi−1 for i ∈ N

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N Auxiliary chain: Ji := ϕτi−1 for i ∈ N Conditionally on τ and J, the excursions {ei(·)}i∈N under Pa

ε,N are

independent and distributed:

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N Auxiliary chain: Ji := ϕτi−1 for i ∈ N Conditionally on τ and J, the excursions {ei(·)}i∈N under Pa

ε,N are

independent and distributed:

◮ for a = p like bridges of the process {Zi}i

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N Auxiliary chain: Ji := ϕτi−1 for i ∈ N Conditionally on τ and J, the excursions {ei(·)}i∈N under Pa

ε,N are

independent and distributed:

◮ for a = p like bridges of the process {Zi}i ◮ for a = w like bridges of the process {Zi}i conditioned to stay

non-negative

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

Excursions and contact set

What happens for ε > 0? The same holds for the excursions. Contact set: τ :=

• i ∈ N : ϕi = 0
• = {τi}i≥0

Excursions: {ei(k)}k := {ϕτi−1+k}0≤k≤τi−τi−1 for i ∈ N Auxiliary chain: Ji := ϕτi−1 for i ∈ N Conditionally on τ and J, the excursions {ei(·)}i∈N under Pa

ε,N are

independent and distributed:

◮ for a = p like bridges of the process {Zi}i ◮ for a = w like bridges of the process {Zi}i conditioned to stay

non-negative Once we know τ, J, the whole field {ϕi}i is reconstructed by pasting independent excursions from {Zi}i (cond. to stay ≥ 0)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34

The Models Free Energy Path Results Proof Integrated random walk

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT) ZNt N3/2

• t∈[0,1]
• condit. on (YN, ZN) = (0, 0) =

⇒

• It
• t∈[0,1]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34

The Models Free Energy Path Results Proof Integrated random walk

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT) ZNt N3/2

• t∈[0,1]
• condit. on (YN, ZN) = (0, 0) =

⇒

• It
• t∈[0,1]

Wetting case: several open issues

◮ Integrated BM conditioned to stay non-negative is studied,

but not its bridge

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34

The Models Free Energy Path Results Proof Integrated random walk

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT) ZNt N3/2

• t∈[0,1]
• condit. on (YN, ZN) = (0, 0) =

⇒

• It
• t∈[0,1]

Wetting case: several open issues

◮ Integrated BM conditioned to stay non-negative is studied,

but not its bridge

◮ Invariance principle seems in any case very difficult

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34

The Models Free Energy Path Results Proof Integrated random walk

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT) ZNt N3/2

• t∈[0,1]
• condit. on (YN, ZN) = (0, 0) =

⇒

• It
• t∈[0,1]

Wetting case: several open issues

◮ Integrated BM conditioned to stay non-negative is studied,

but not its bridge

◮ Invariance principle seems in any case very difficult

Entropic repulsion

P

• Z1 ≥ 0 , . . . , ZN ≥ 0
• ≈ ?

P

• Z1 ≥ 0 , . . . , ZN ≥ 0
• YN = 0, ZN = 0
• ≈ ?

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34

The Models Free Energy Path Results Proof Integrated random walk

The law of the excursions

Pinning case: good control (Donsker’s inv. pr. + LLT) ZNt N3/2

• t∈[0,1]
• condit. on (YN, ZN) = (0, 0) =

⇒

• It
• t∈[0,1]

Wetting case: several open issues

◮ Integrated BM conditioned to stay non-negative is studied,

but not its bridge

◮ Invariance principle seems in any case very difficult

Entropic repulsion

P

• Z1 ≥ 0 , . . . , ZN ≥ 0
• ≈ N−1/4

[Sinai (SRW)] P

• Z1 ≥ 0 , . . . , ZN ≥ 0
• YN = 0, ZN = 0
• ≈ N−1/2

[conj.]

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34

The Models Free Energy Path Results Proof Markov renewal theory

Markov renewal processes

Given a (sub-)probability kernel Kx,dy(n):

• y∈R
• n∈N

Kx,dy(n) = c ≤ 1 , ∀x ∈ R we build the Markov renewal process τ with modulating chain J: P

• τi+1 − τi = n , Ji+1 ∈ dy
• Ji = x
• := Kx,dy(n)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 31 / 34

The Models Free Energy Path Results Proof Markov renewal theory

Markov renewal processes

Given a (sub-)probability kernel Kx,dy(n):

• y∈R
• n∈N

Kx,dy(n) = c ≤ 1 , ∀x ∈ R we build the Markov renewal process τ with modulating chain J: P

• τi+1 − τi = n , Ji+1 ∈ dy
• Ji = x
• := Kx,dy(n)

The law of (τ, J) conditionally on {N, N + 1} ⊆ τ is P

• τi = ti , Ji ∈ dyi
• {N, N + 1} ⊆ τ
• =

1 CN

• i

Kyi−1,dyi(ti − ti−1) with CN = P({N, N + 1} ⊆ τ).

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 31 / 34

The Models Free Energy Path Results Proof Markov renewal theory

The law of the contact set

Consider the following kernels: for n ∈ N and x, y ∈ R Gp

x,dy(n) := ε Px

• Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

Gw

x,dy(n) := ε Px

• Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34

The Models Free Energy Path Results Proof Markov renewal theory

The law of the contact set

Consider the following kernels: for n ∈ N and x, y ∈ R Gp

x,dy(n) := ε Px

• Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

Gw

x,dy(n) := ε Px

• Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

The law of (τ, J) is given by Pa

ε,N

• τi = ti , Ji ∈ dyi , i ≤ k
• =

1 Za

ε,N k

• i=1

Ga

yi−1,dyi(ti − ti−1)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34

The Models Free Energy Path Results Proof Markov renewal theory

The law of the contact set

Consider the following kernels: for n ∈ N and x, y ∈ R Gp

x,dy(n) := ε Px

• Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

Gw

x,dy(n) := ε Px

• Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
• dz
• z=0

The law of (τ, J) is given by Pa

ε,N

• τi = ti , Ji ∈ dyi , i ≤ k
• =

1 Za

ε,N k

• i=1

Ga

yi−1,dyi(ti − ti−1)

Reminds of Markov renewal theory

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34

The Models Free Energy Path Results Proof Markov renewal theory

Markov renewal processes

We exploit the invariance properties: for every F, v(y) Pa

ε,N

• τi = ti , Ji ∈ dyi , i ≤ k
• =

eFN Za

ε,N k

• i=1

Ga

yi−1,dyi(ti − ti−1) e−F(ti−ti−1)

v(yi) v(yi−1)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 33 / 34

The Models Free Energy Path Results Proof Markov renewal theory

Markov renewal processes

We exploit the invariance properties: for every F, v(y) Pa

ε,N

• τi = ti , Ji ∈ dyi , i ≤ k
• =

eFN Za

ε,N k

• i=1

Ga

yi−1,dyi(ti − ti−1) e−F(ti−ti−1)

v(yi) v(yi−1) If we determine F, v(·) such that Kx,dy(n) := Ga

x,dy(n) e−F·n v(y)

v(x) is a probability kernel, we have the crucial relation Pa

ε,N

• τi = ti , Ji ∈ dyi
• = P
• τi = ti , Ji ∈ dyi
• {N, N + 1} ⊆ τ
• Francesco Caravenna

(1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 33 / 34

The Models Free Energy Path Results Proof Markov renewal theory

A Perron-Frobenius problem

It turns out that:

◮ F is the solution of the equation

spectral radius of

• n∈N

Ga

x,dy(n) e−F·n

• x,y

= 1 when such a solution exists and F = 0 otherwise.

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 34 / 34

The Models Free Energy Path Results Proof Markov renewal theory

A Perron-Frobenius problem

It turns out that:

◮ F is the solution of the equation

spectral radius of

• n∈N

Ga

x,dy(n) e−F·n

• x,y

= 1 when such a solution exists and F = 0 otherwise.

◮ In fact F = Fa(ε) is the free energy

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 34 / 34

The Models Free Energy Path Results Proof Markov renewal theory

A Perron-Frobenius problem

It turns out that:

◮ F is the solution of the equation

spectral radius of

• n∈N

Ga

x,dy(n) e−F·n

• x,y

= 1 when such a solution exists and F = 0 otherwise.

◮ In fact F = Fa(ε) is the free energy ◮ v(·) is the principal eigenfunction:

• y∈R
• n∈N

Ga

x,dy(n) e−F·n

• x,y

v(y) = v(x)

Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 34 / 34