A Polymer in a Multi-Interface Medium Francesco Caravenna - - PowerPoint PPT Presentation

Introduction The model The proof

A Polymer in a Multi-Interface Medium

Francesco Caravenna

Universit` a degli Studi di Padova

LPMA ∼ December 8, 2009

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 1 / 26

Introduction The model The proof

References

◮ [CP1] F. Caravenna and N. P´

etr´ elis A polymer in a multi-interface medium AAP (2009)

◮ [CP2] F. Caravenna and N. P´

etr´ elis Depinning of a polymer in a multi-interface medium EJP (2009)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 2 / 26

Introduction The model The proof

Outline

• 1. Introduction and motivations

Polymer models

• 2. The model and the main results

Definition of the model The free energy Path results

• 3. Techniques and ideas from the proof

Some heuristics A renewal theory approach

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 3 / 26

Introduction The model The proof

Outline

• 1. Introduction and motivations

Polymer models

• 2. The model and the main results

Definition of the model The free energy Path results

• 3. Techniques and ideas from the proof

Some heuristics A renewal theory approach

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 4 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Oil Water Interface

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Copolymer + + + + + + + + + + + + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + _ _ Oil Water Interface +

◮ Copolymer interaction with the solvents

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Pinning Oil Water Interface

◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Pinning Copolymer + + + + + + + + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + + _ _ Oil Water Interface + + + + +

◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Pinning Copolymer + + + + + + + + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + + _ _ Oil Water Interface + + + + +

◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered

Localization vs. delocalization? Phase transitions?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Copolymer and pinning at a single interface

A polymer interacting with two solvents and with the interface that separates them:

Pinning Copolymer + + + + + + + + + + + + + + + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ + + + + _ _ Oil Water Interface + + + + +

◮ Copolymer interaction with the solvents ◮ Pinning interaction with the interface ◮ Both interactions may be inhomogeneous and disordered

Localization vs. delocalization? Phase transitions? Recent results: very good comprehension (survey: [Giacomin ’07])

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 5 / 26

Introduction The model The proof Polymer models

Multi-interface media

More general environments: a multi-interface medium

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26

Introduction The model The proof Polymer models

Multi-interface media

More general environments: a multi-interface medium

T + + + + + + + + + + + + +++ + + + + + + + + _ __ __ _ _ _ __ _ _ _ _ _ _ __ _ _ _ _ _ _ + _ + Water Water Oil _ _ _ +

◮ [den Hollander & W¨

uthrich JSP 04]: Copolymer interaction. Path results for log log N ≪ TN ≪ log N

(N = polymer size)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26

Introduction The model The proof Polymer models

Multi-interface media

More general environments: a multi-interface medium

T

◮ [den Hollander & W¨

uthrich JSP 04]: Copolymer interaction. Path results for log log N ≪ TN ≪ log N

(N = polymer size)

◮ We focus on the pinning case. Homogeneous interaction

(attractive or repulsive), general TN − → Path behavior ?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 6 / 26

Introduction The model The proof Polymer models

Polymer in a slit

Recent physical literature: Polymer confined between two walls and interacting with them

◮ [Brak et al.; J Phys A 2005] ◮ [Martin et al.; J Phys A 2007] ◮ [Owkzarek et al.; J Phys A 2008] T δ δ

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 7 / 26

Introduction The model The proof Polymer models

Polymer in a slit

Recent physical literature: Polymer confined between two walls and interacting with them

◮ [Brak et al.; J Phys A 2005] ◮ [Martin et al.; J Phys A 2007] ◮ [Owkzarek et al.; J Phys A 2008] T δ δ

Attraction/repulsion of interfaces by polymers

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 7 / 26

Introduction The model The proof Polymer models

Polymer in a slit

T T 2T 3T −T δ δ δ δ δ δ + log 2 δ + log 2

ΦT

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 8 / 26

Introduction The model The proof

Outline

• 1. Introduction and motivations

Polymer models

• 2. The model and the main results

Definition of the model The free energy Path results

• 3. Techniques and ideas from the proof

Some heuristics A renewal theory approach

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 9 / 26

Introduction The model The proof Definition of the model

Definition

Recall the situation we want to model

T

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26

Introduction The model The proof Definition of the model

Definition

Recall the situation we want to model

T

Polymer configurations ← → Trajectories of a random process

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26

Introduction The model The proof Definition of the model

Definition

Recall the situation we want to model

T

Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure PT

N,δ on paths

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26

Introduction The model The proof Definition of the model

Definition

Recall the situation we want to model

T

Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure PT

N,δ on paths ◮ (1 + 1)-dimensionale model: {(i, Si)}i≥0

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26

Introduction The model The proof Definition of the model

Definition

Recall the situation we want to model

T

Polymer configurations ← → Trajectories of a random process Random polymer model: probability measure PT

N,δ on paths ◮ (1 + 1)-dimensionale model: {(i, Si)}i≥0 ◮ PT N,δ absolutely continuous w.r.t. SRW {Si}i≥0

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 10 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2. ◮ Polymer size N ∈ 2N (number of monomers)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2. ◮ Polymer size N ∈ 2N (number of monomers) ◮ Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2. ◮ Polymer size N ∈ 2N (number of monomers) ◮ Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive) ◮ Interface distance T ∈ 2N (interfaces ≡ TZ)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2. ◮ Polymer size N ∈ 2N (number of monomers) ◮ Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive) ◮ Interface distance T ∈ 2N (interfaces ≡ TZ)

Polymer measure:

dPT

N,δ

dP (S) := 1 Z T

N,δ

exp

• HT

N,δ(S)

• Francesco Caravenna

A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Ingredients of PT

N,δ: ◮ Simple symmetric random walk S = {Sn}n≥0 on Z:

S0 := 0 , Sn := X1 + . . . + Xn , with {Xi}i i.i.d. and P(Xi = +1) = P(Xi = −1) = 1

2. ◮ Polymer size N ∈ 2N (number of monomers) ◮ Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive) ◮ Interface distance T ∈ 2N (interfaces ≡ TZ)

Polymer measure:

dPT

N,δ

dP (S) := 1 Z T

N,δ

exp

• HT

N,δ(S)

• =

1 Z T

N,δ

exp

• δ

N

• i=1

1{Si ∈ TZ}

• Francesco Caravenna

A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Definition

Sn N T

Polymer measure:

dPT

N,δ

dP (S) := 1 Z T

N,δ

exp

• HT

N,δ(S)

• =

1 Z T

N,δ

exp

• δ

N

• i=1

1{Si ∈ TZ}

• Francesco Caravenna

A Polymer in a Multi-Interface Medium December 8, 2009 11 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Some questions on PTN

N,δ for large N ◮ What is the typical size of SN?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Some questions on PTN

N,δ for large N ◮ What is the typical size of SN? ◮ How many monomers touch the interfaces?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Some questions on PTN

N,δ for large N ◮ What is the typical size of SN? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Some questions on PTN

N,δ for large N ◮ What is the typical size of SN? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited? ◮ Interplay between the parameters δ and T = {TN}N?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof Definition of the model

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions Interested in the properties of PT

N,δ as N → ∞

(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN

Some questions on PTN

N,δ for large N ◮ What is the typical size of SN? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited? ◮ Interplay between the parameters δ and T = {TN}N?

Penalisation of the simple random walk

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 12 / 26

Introduction The model The proof The free energy

The free energy

The free energy φ(δ, {Tn}n) encodes the exponential asymptotic behavior of the normalization constant Z TN

N,δ (partition function)

φ(δ, {Tn}n) := lim

N→∞

1 N log Z TN

N,δ

(super-additivity)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 13 / 26

Introduction The model The proof The free energy

The free energy

The free energy φ(δ, {Tn}n) encodes the exponential asymptotic behavior of the normalization constant Z TN

N,δ (partition function)

φ(δ, {Tn}n) := lim

N→∞

1 N log Z TN

N,δ

(super-additivity)

Z TN

N,δ := E

• exp
• δ

N

• i=1

1{Si ∈ TZ}

• = E
• exp
• δLN
• LN := #{i ≤ N : Si ∈ TZ} number of visits to the interfaces

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 13 / 26

Introduction The model The proof The free energy

The free energy

The free energy φ(δ, {Tn}n) encodes the exponential asymptotic behavior of the normalization constant Z TN

N,δ (partition function)

φ(δ, {Tn}n) := lim

N→∞

1 N log Z TN

N,δ

(super-additivity)

Z TN

N,δ := E

• exp
• δ

N

• i=1

1{Si ∈ TZ}

• = E
• exp
• δLN
• LN := #{i ≤ N : Si ∈ TZ} number of visits to the interfaces

φ is a generating function

φ′(δ, {Tn}n) = lim

N→∞ ETN N,δ

LN N

• LN ∼ φ′(δ, {Tn}n) · N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 13 / 26

Introduction The model The proof The free energy

The free energy

The free energy φ(δ, {Tn}n) encodes the exponential asymptotic behavior of the normalization constant Z TN

N,δ (partition function)

φ(δ, {Tn}n) := lim

N→∞

1 N log Z TN

N,δ

(super-additivity)

Z TN

N,δ := E

• exp
• δ

N

• i=1

1{Si ∈ TZ}

• = E
• exp
• δLN
• LN := #{i ≤ N : Si ∈ TZ} number of visits to the interfaces

φ is a generating function

φ′(δ, {Tn}n) = lim

N→∞ ETN N,δ

LN N

• LN ∼ φ′(δ, {Tn}n) · N

φ(δ, {Tn}n) non-analytic at δ ← → phase transition at δ

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 13 / 26

Introduction The model The proof The free energy

The free energy: characterization and results

We may assume that limN→∞ TN =: T∞ ∈ 2N ∪ {+∞} exists

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 14 / 26

Introduction The model The proof The free energy

The free energy: characterization and results

We may assume that limN→∞ TN =: T∞ ∈ 2N ∪ {+∞} exists Let τ T

1 := inf

• n > 0 : Sn ∈ {±T, 0}
• and QT(λ) := E
• e−λτ T

1

(Also T = ∞: first return to zero of the SSRW)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 14 / 26

Introduction The model The proof The free energy

The free energy: characterization and results

We may assume that limN→∞ TN =: T∞ ∈ 2N ∪ {+∞} exists Let τ T

1 := inf

• n > 0 : Sn ∈ {±T, 0}
• and QT(λ) := E
• e−λτ T

1

(Also T = ∞: first return to zero of the SSRW)

Theorem ([CP1])

φ(δ, {Tn}n) = φ(δ, T∞) =

• QT∞

−1(e−δ) if T∞ < +∞

• Q∞

−1(e−δ ∧ 1) if T∞ = +∞

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 14 / 26

Introduction The model The proof The free energy

The free energy: characterization and results

We may assume that limN→∞ TN =: T∞ ∈ 2N ∪ {+∞} exists Let τ T

1 := inf

• n > 0 : Sn ∈ {±T, 0}
• and QT(λ) := E
• e−λτ T

1

(Also T = ∞: first return to zero of the SSRW)

Theorem ([CP1])

φ(δ, {Tn}n) = φ(δ, T∞) =

• QT∞

−1(e−δ) if T∞ < +∞

• Q∞

−1(e−δ ∧ 1) if T∞ = +∞

If TN → T∞ < ∞ (same as TN ≡ T∞)

◮ φ(δ, T∞) is analytic on R: no phase transitions

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 14 / 26

Introduction The model The proof The free energy

The free energy: characterization and results

We may assume that limN→∞ TN =: T∞ ∈ 2N ∪ {+∞} exists Let τ T

1 := inf

• n > 0 : Sn ∈ {±T, 0}
• and QT(λ) := E
• e−λτ T

1

(Also T = ∞: first return to zero of the SSRW)

Theorem ([CP1])

φ(δ, {Tn}n) = φ(δ, T∞) =

• QT∞

−1(e−δ) if T∞ < +∞

• Q∞

−1(e−δ ∧ 1) if T∞ = +∞

If TN → T∞ < ∞ (same as TN ≡ T∞)

◮ φ(δ, T∞) is analytic on R: no phase transitions ◮ φ′(δ, T∞) > 0 for every δ ∈ R: positive density of contacts

LN ∼ φ′(δ, T∞) · N (diffusive scaling of SN)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 14 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

◮ If δ > 0 then φ′(δ, ∞) > 0 −

→ LN ∼ φ′(δ, ∞) · N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

◮ If δ > 0 then φ′(δ, ∞) > 0 −

→ LN ∼ φ′(δ, ∞) · N

◮ Every {Tn}n → ∞ yields the same free energy as if Tn ≡ ∞

(homogeneous pinning model) − → same density of visits

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

◮ If δ > 0 then φ′(δ, ∞) > 0 −

→ LN ∼ φ′(δ, ∞) · N

◮ Every {Tn}n → ∞ yields the same free energy as if Tn ≡ ∞

(homogeneous pinning model) − → same density of visits

◮ Same path behavior? NO!

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

◮ If δ > 0 then φ′(δ, ∞) > 0 −

→ LN ∼ φ′(δ, ∞) · N

◮ Every {Tn}n → ∞ yields the same free energy as if Tn ≡ ∞

(homogeneous pinning model) − → same density of visits

◮ Same path behavior? NO! ◮ If δ < 0 then Z TN N,δ = exp(o(N)).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof The free energy

The free energy: results

If TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) = φ′(δ, ∞) ≡ 0 −

→ LN = o(N)

◮ If δ > 0 then φ′(δ, ∞) > 0 −

→ LN ∼ φ′(δ, ∞) · N

◮ Every {Tn}n → ∞ yields the same free energy as if Tn ≡ ∞

(homogeneous pinning model) − → same density of visits

◮ Same path behavior? NO! ◮ If δ < 0 then Z TN N,δ = exp(o(N)). In fact

Z TN

N,δ ≈ (const.)

N3/2 f N T 2

N

• g

N T 3

N

• ,

improving known results for the polymer in a slit.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 15 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N) SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N)

◮ If TN − 1 cδ log N → ζ ∈ R then SN ≍ TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N)

◮ If TN − 1 cδ log N → ζ ∈ R then SN ≍ TN

SN TN = ⇒ SΓ Γ ∼ Poisson(f (δ, ζ))

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N)

◮ If TN − 1 cδ log N → ζ ∈ R then SN ≍ TN ◮ If TN − 1 cδ log N → + ∞ then SN ≍ O(1)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

Assume δ > 0 and TN → ∞. The polymer visits the interfaces a positive fraction of times: LN ∼ φ′(δ, ∞) · N

(φ′(δ, ∞) > 0)

What is the typical size of SN? Not.: SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Theorem ([CP1])

For every δ > 0 there exists cδ > 0 such that under PTN

N,δ: ◮ If TN − 1 cδ log N → − ∞ then SN ≍ (e− cδ

2 TN TN)

√ N (note that TN ≪ SN ≪ √ N)

◮ If TN − 1 cδ log N → ζ ∈ R then SN ≍ TN ◮ If TN − 1 cδ log N → + ∞ then SN ≍ O(1)

lim

L→∞ sup N∈2N

PTN

N,δ

• |SN| > L
• = 0

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 16 / 26

Introduction The model The proof Path results

Path results: the attractive case δ > 0

SN

√ N

1 cδ log N 1 cδ log N

O(1) 1

TN

• Sub-diffusive scaling (TN → ∞)
• Transition at TN ≈ log N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 17 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Theorem ([CP2])

For every δ < 0 we have under PTN

N,δ: ◮ If TN ≪ N1/3 then SN ≍

• N/TN ≫ TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Theorem ([CP2])

For every δ < 0 we have under PTN

N,δ: ◮ If TN ≪ N1/3 then SN ≍

• N/TN ≫ TN

c1 P

• a < Z ≤ b
• ≤ PTN

N,δ

• a <

SN Cδ

• N

TN

≤ b

• ≤ c2 P
• a < Z ≤ b
• with Z ∼ N(0, 1)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Theorem ([CP2])

For every δ < 0 we have under PTN

N,δ: ◮ If TN ≪ N1/3 then SN ≍

• N/TN ≫ TN

◮ If (const.)N1/3 ≤ TN ≤ (const.)

√ N then SN ≍ TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Theorem ([CP2])

For every δ < 0 we have under PTN

N,δ: ◮ If TN ≪ N1/3 then SN ≍

• N/TN ≫ TN

◮ If (const.)N1/3 ≤ TN ≤ (const.)

√ N then SN ≍ TN

◮ If TN ≫

√ N then SN ≍ √ N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

Assume δ < 0 and TN → ∞. The polymer visits the interfaces a null fraction of times: LN = o(N). What is the typical size of SN?

Theorem ([CP2])

For every δ < 0 we have under PTN

N,δ: ◮ If TN ≪ N1/3 then SN ≍

• N/TN ≫ TN

◮ If (const.)N1/3 ≤ TN ≤ (const.)

√ N then SN ≍ TN

◮ If TN ≫

√ N then SN ≍ √ N

◮ If TN ∼ (const.)N1/3 −

→ O(1) visited interfaces.

◮ If TN ≫ N1/3 −

→ 1 visited interface, LN = O(1).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 18 / 26

Introduction The model The proof Path results

Path results: the repulsive case δ < 0

SN

√ N √ N N1/3 N1/3 1

TN

• Sub-diffusive if 1 ≪ TN ≪

√ N

• Transitions TN ≈ N1/3,

√ N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 19 / 26

Introduction The model The proof

Outline

• 1. Introduction and motivations

Polymer models

• 2. The model and the main results

Definition of the model The free energy Path results

• 3. Techniques and ideas from the proof

Some heuristics A renewal theory approach

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 20 / 26

Introduction The model The proof Some heuristics

An heuristic view: the attractive case δ > 0

Consider the simple homogeneous pinning model (one interface at zero) − → for δ > 0 it is localized at the interface

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 21 / 26

Introduction The model The proof Some heuristics

An heuristic view: the attractive case δ > 0

Consider the simple homogeneous pinning model (one interface at zero) − → for δ > 0 it is localized at the interface

N max |Si| ≈ log N SN = O(1)

max

0≤i≤N |Si| ≍ log N

SN = O(1)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 21 / 26

Introduction The model The proof Some heuristics

An heuristic view: the attractive case δ > 0

Consider the simple homogeneous pinning model (one interface at zero) − → for δ > 0 it is localized at the interface

N TN max |Si| ≈ log N SN = O(1)

max

0≤i≤N |Si| ≍ log N

SN = O(1)

◮ If TN ≫ log N nothing changes: polymer localized at zero

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 21 / 26

Introduction The model The proof Some heuristics

An heuristic view: the attractive case δ > 0

Consider the simple homogeneous pinning model (one interface at zero) − → for δ > 0 it is localized at the interface

N TN

max

0≤i≤N |Si| ≍ log N

SN = O(1)

◮ If TN ≫ log N nothing changes: polymer localized at zero ◮ If TN ≪ log N it is worth to visit different interfaces:

sub-diffusive behavior

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 21 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T2

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T3

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Under the polymer measure PT

N,δ with δ < 0 ˆ

τ T

1 ≈ T 3 (repulsion)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T3

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Under the polymer measure PT

N,δ with δ < 0 ˆ

τ T

1 ≈ T 3 (repulsion) ◮ if T 3 N ≪ N infinitely many interfaces: SN ≈ TN

• N

T 3

N =

• N

TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T3

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Under the polymer measure PT

N,δ with δ < 0 ˆ

τ T

1 ≈ T 3 (repulsion) ◮ if T 3 N ≪ N infinitely many interfaces: SN ≈ TN

• N

T 3

N =

• N

TN ◮ if T 3 N ≫ N confinement: |Si| < TN for all i ≤ N:

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T3

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Under the polymer measure PT

N,δ with δ < 0 ˆ

τ T

1 ≈ T 3 (repulsion) ◮ if T 3 N ≪ N infinitely many interfaces: SN ≈ TN

• N

T 3

N =

• N

TN ◮ if T 3 N ≫ N confinement: |Si| < TN for all i ≤ N:

• SN ≈ TN (if TN ≤

√ N)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof Some heuristics

An heuristic view: the repulsive case δ < 0

How long is the time ˆ τ T

1 to reach level ±T?

N T ˆ τ T

1 ≈ T3

Under the simple random walk law ˆ τ T

1 ≈ T 2 (diffusivity)

Under the polymer measure PT

N,δ with δ < 0 ˆ

τ T

1 ≈ T 3 (repulsion) ◮ if T 3 N ≪ N infinitely many interfaces: SN ≈ TN

• N

T 3

N =

• N

TN ◮ if T 3 N ≫ N confinement: |Si| < TN for all i ≤ N:

• SN ≈ TN (if TN ≤

√ N)

• SN ≈

√ N (if TN ≫ √ N)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 22 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Let τ T

1 , τ T 2 , τ T 3 . . . be the points at which Sn visits an interface

τ T

k+1 := inf

• n > τ T

k : Sn − Sτ T

k ∈ {−T, 0, T}

• (T is fixed)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 23 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Let τ T

1 , τ T 2 , τ T 3 . . . be the points at which Sn visits an interface

τ T

k+1 := inf

• n > τ T

k : Sn − Sτ T

k ∈ {−T, 0, T}

• (T is fixed)

Under the simple random walk law {τ T

n }n∈N is a classical renewal

process with explicit law qT(n) := P(τ T

1 = n) ≈

1 (min{n, T 2})3/2 e− π2

2T2 n Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 23 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Let τ T

1 , τ T 2 , τ T 3 . . . be the points at which Sn visits an interface

τ T

k+1 := inf

• n > τ T

k : Sn − Sτ T

k ∈ {−T, 0, T}

• (T is fixed)

Under the simple random walk law {τ T

n }n∈N is a classical renewal

process with explicit law qT(n) := P(τ T

1 = n) ≈

1 (min{n, T 2})3/2 e− π2

2T2 n

Note that under P τ T

1

≈      O(1) with probab. 1 − 1

T

• qT(n) ≈

1 n3/2

• Francesco Caravenna

A Polymer in a Multi-Interface Medium December 8, 2009 23 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Let τ T

1 , τ T 2 , τ T 3 . . . be the points at which Sn visits an interface

τ T

k+1 := inf

• n > τ T

k : Sn − Sτ T

k ∈ {−T, 0, T}

• (T is fixed)

Under the simple random walk law {τ T

n }n∈N is a classical renewal

process with explicit law qT(n) := P(τ T

1 = n) ≈

1 (min{n, T 2})3/2 e− π2

2T2 n

Note that under P τ T

1

≈      O(1) with probab. 1 − 1

T

• qT(n) ≈

1 n3/2

• O(T 2) with probab. 1

T

• qT(n) ≈

1 T 3 e− π2

2T2 n Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 23 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Under the polymer measure PT

N,δ the process {τ T n }n∈N is not even

time-homogeneous . . .

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 24 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Under the polymer measure PT

N,δ the process {τ T n }n∈N is not even

time-homogeneous . . . however for large N it is nearly a renewal process with a different law Pδ,T: for both δ > 0 and δ < 0 Kδ,T(n) := Pδ,T(τ T

1 = n) = eδ P(τ T 1 = n) e−φ(δ,T)n

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 24 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Under the polymer measure PT

N,δ the process {τ T n }n∈N is not even

time-homogeneous . . . however for large N it is nearly a renewal process with a different law Pδ,T: for both δ > 0 and δ < 0 Kδ,T(n) := Pδ,T(τ T

1 = n) = eδ P(τ T 1 = n) e−φ(δ,T)n

For δ > 0, we have φ(δ, T) → φ(δ, ∞) > 0 as T → ∞, hence Kδ,T(n) ≈ e−φ(δ,∞)n

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 24 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Under the polymer measure PT

N,δ the process {τ T n }n∈N is not even

time-homogeneous . . . however for large N it is nearly a renewal process with a different law Pδ,T: for both δ > 0 and δ < 0 Kδ,T(n) := Pδ,T(τ T

1 = n) = eδ P(τ T 1 = n) e−φ(δ,T)n

For δ > 0, we have φ(δ, T) → φ(δ, ∞) > 0 as T → ∞, hence Kδ,T(n) ≈ e−φ(δ,∞)n For δ < 0, we have φ(δ, T) ≈ − π2

2T 2 + Cδ T 3 as T → ∞, hence

τ T

1

≈      O(1) with probab. eδ

• Kδ,T(n) ≈

1 n3/2

• Francesco Caravenna

A Polymer in a Multi-Interface Medium December 8, 2009 24 / 26

Introduction The model The proof A renewal theory approach

A renewal theory approach

Under the polymer measure PT

N,δ the process {τ T n }n∈N is not even

time-homogeneous . . . however for large N it is nearly a renewal process with a different law Pδ,T: for both δ > 0 and δ < 0 Kδ,T(n) := Pδ,T(τ T

1 = n) = eδ P(τ T 1 = n) e−φ(δ,T)n

For δ > 0, we have φ(δ, T) → φ(δ, ∞) > 0 as T → ∞, hence Kδ,T(n) ≈ e−φ(δ,∞)n For δ < 0, we have φ(δ, T) ≈ − π2

2T 2 + Cδ T 3 as T → ∞, hence

τ T

1

≈      O(1) with probab. eδ

• Kδ,T(n) ≈

1 n3/2

• O(T 3) with probab. 1 − eδ
• Kδ,T(n) ≈

1 T 3 e− Cδ

T3 n Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 24 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

• 1. Study τ TN ∩ [0, N] under Pδ,TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

• 1. Study τ TN ∩ [0, N] under Pδ,TN
• 2. Transfer the results to Pδ,TN( · | N ∈ τ TN) (hard part)

In this way we control PTN

N,δ( · | N ∈ τ TN)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

• 1. Study τ TN ∩ [0, N] under Pδ,TN
• 2. Transfer the results to Pδ,TN( · | N ∈ τ TN) (hard part)

In this way we control PTN

N,δ( · | N ∈ τ TN)

• 3. Remove the conditioning on {N ∈ τ TN}

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

• 1. Study τ TN ∩ [0, N] under Pδ,TN
• 2. Transfer the results to Pδ,TN( · | N ∈ τ TN) (hard part)

In this way we control PTN

N,δ( · | N ∈ τ TN)

• 3. Remove the conditioning on {N ∈ τ TN}

◮ Good estimates on qT(n) and on the free energy φ(δ, T)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Strategy of the proof

For fixed T, the law of τ T ∩ [0, N] = {τ T

1 , . . . , τ T LN} is the same

under PT

N,δ

• ·
• N ∈ τ T

and Pδ,T

• ·
• N ∈ τ T
• Pδ,T does not depend explicitly on N
• PT

N,δ =

⇒ Pδ,T However we want to study PTN

N,δ with TN varying with N

• 1. Study τ TN ∩ [0, N] under Pδ,TN
• 2. Transfer the results to Pδ,TN( · | N ∈ τ TN) (hard part)

In this way we control PTN

N,δ( · | N ∈ τ TN)

• 3. Remove the conditioning on {N ∈ τ TN}

◮ Good estimates on qT(n) and on the free energy φ(δ, T) ◮ Uniform renewal theorems

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 25 / 26

Introduction The model The proof A renewal theory approach

Merci.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 8, 2009 26 / 26