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

SLIDE 1

Introduction The Model Free Energy Path Behavior The Proof

A Polymer in a Multi-Interface Medium

Francesco Caravenna

Universit` a degli Studi di Milano-Bicocca

Joint work with Nicolas P´ etr´ elis (Nantes)

Universit¨ at Bonn ∼ December 16, 2010

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 2

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 3

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 4

Introduction The Model Free Energy Path Behavior The Proof

A polymer in a multi-interface medium

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 5

Introduction The Model Free Energy Path Behavior The Proof

A polymer in a multi-interface medium

Single interface case is well understood.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 6

Introduction The Model Free Energy Path Behavior The Proof

A polymer in a multi-interface medium

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

Single interface case is well understood. Copolymer interaction [den Hollander & W¨ uthrich JSP 04] Some path results for log log N ≪TN ≪log N (N = polymer size)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 7

Introduction The Model Free Energy Path Behavior The Proof

A polymer in a multi-interface medium

T

Single interface case is well understood. Copolymer interaction [den Hollander & W¨ uthrich JSP 04] Some path results for log log N ≪TN ≪log N (N = polymer size) Focus on (homogeneous, attractive/repulsive) pinning interaction. Path behavior? Interplay between N and T = TN?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 8

Introduction The Model Free Energy Path Behavior The Proof

Stabilization of colloidal dispersions

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 9

Introduction The Model Free Energy Path Behavior The Proof

Stabilization of colloidal dispersions

The addition of polymers into a colloid can prevent the aggregation

f droplets via entropic repulsion (steric stabilization of the colloid)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 10

Introduction The Model Free Energy Path Behavior The Proof

Stabilization of colloidal dispersions

The addition of polymers into a colloid can prevent the aggregation

f droplets via entropic repulsion (steric stabilization of the colloid)

Polymer confined between two walls and interacting with them: (polymer in a slit)

T δ δ

Physics literature: [Brak et al. 2005], [Owkzarek et al. 2008], . . .

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 11

Introduction The Model Free Energy Path Behavior The Proof

Multi-interface medium vs. 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 16, 2010

SLIDE 12

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 13

Introduction The Model Free Energy Path Behavior The Proof

Definition

Recall the situation we want to model

T

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 14

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 15

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 16

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 17

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 18

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 19

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 20

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 21

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 22

Introduction The Model Free Energy Path Behavior The Proof

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

δLN(S)
Francesco Caravenna

A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 23

Introduction The Model Free Energy Path Behavior The Proof

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

δLN(S)
=

1 Z T

N,δ

exp

δ

N

i=1

1{Si ∈ TZ}

Francesco Caravenna

A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 24

Introduction The Model Free Energy Path Behavior The Proof

Definition

Sn N T

Polymer measure:

dPT

N,δ

dP (S) := 1 Z T

N,δ

exp

δLN(S)
=

1 Z T

N,δ

exp

δ

N

i=1

1{Si ∈ TZ}

Francesco Caravenna

A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 25

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 26

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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 16, 2010

SLIDE 27

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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 16, 2010

SLIDE 28

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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 16, 2010

SLIDE 29

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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 16, 2010

SLIDE 30

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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 16, 2010

SLIDE 31

Introduction The Model Free Energy Path Behavior The Proof

Some questions

The law PT

N,δ describes the statistical distribution of the

configurations of the polymer, given the external conditions We are 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?

Penalization of the simple random walk

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 32

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 33

Introduction The Model Free Energy Path Behavior The Proof

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)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 34

Introduction The Model Free Energy Path Behavior The Proof

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 + . . . )

= lim

N→∞

1 N log E

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

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 35

Introduction The Model Free Energy Path Behavior The Proof

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 + . . . )

= lim

N→∞

1 N log E

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

φ is a generating function:

∂ ∂δφ(δ, {Tn}n) = lim

N→∞ ETN N,δ

LN N

LN ∼ ∂φ

∂δ · N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 36

Introduction The Model Free Energy Path Behavior The Proof

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 + . . . )

= lim

N→∞

1 N log E

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

φ is a generating function:

∂ ∂δφ(δ, {Tn}n) = lim

N→∞ ETN N,δ

LN N

LN ∼ ∂φ

∂δ · N

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

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 37

Introduction The Model Free Energy Path Behavior The Proof

The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

◮ either TN ≡ T < ∞ is constant for N large ◮ or TN → ∞ as N → ∞ (T = ∞)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 38

Introduction The Model Free Energy Path Behavior The Proof

The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

◮ either TN ≡ T < ∞ is constant for N large ◮ or TN → ∞ as N → ∞ (T = ∞)

Let τ T

1 := inf

n > 0 : Sn ∈ {±T, 0}
and set

QT(λ) := E

e−λτ T

1 Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 39

Introduction The Model Free Energy Path Behavior The Proof

The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

◮ either TN ≡ T < ∞ is constant for N large ◮ or TN → ∞ as N → ∞ (T = ∞)

Let τ T

1 := inf

n > 0 : Sn ∈ {±T, 0}
and set

QT(λ) := E

e−λτ T

1

◮ if T < ∞, QT : (−λT, ∞) → (0, ∞)

(λT ∼ − π2

2T 2 )

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 40

Introduction The Model Free Energy Path Behavior The Proof

The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

◮ either TN ≡ T < ∞ is constant for N large ◮ or TN → ∞ as N → ∞ (T = ∞)

Let τ T

1 := inf

n > 0 : Sn ∈ {±T, 0}
and set

QT(λ) := E

e−λτ T

1

◮ if T < ∞, QT : (−λT, ∞) → (0, ∞)

(λT ∼ − π2

2T 2 )

◮ if T = ∞, Q∞ : [0, ∞) → (0, 1]

(first return to zero of SRW)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 41

Introduction The Model Free Energy Path Behavior The Proof

The free energy: characterization

We assume that TN → T ∈ 2N ∪ {∞}, i.e.,

◮ either TN ≡ T < ∞ is constant for N large ◮ or TN → ∞ as N → ∞ (T = ∞)

Let τ T

1 := inf

n > 0 : Sn ∈ {±T, 0}
and set

QT(λ) := E

e−λτ T

1

◮ if T < ∞, QT : (−λT, ∞) → (0, ∞)

(λT ∼ − π2

2T 2 )

◮ if T = ∞, Q∞ : [0, ∞) → (0, 1]

(first return to zero of SRW)

Theorem ([CP1]). Let TN → T.

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 16, 2010

SLIDE 42

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

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

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 43

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 44

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 45

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Case TN → ∞

◮ Phase transition (only) at δ = 0

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 46

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Case TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) ≡ ∂

∂δφ(δ, ∞) ≡ 0 LN = o(N)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 47

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Case TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) ≡ ∂

∂δφ(δ, ∞) ≡ 0 LN = o(N)

◮ If δ > 0 then

∂ ∂δφ(δ, ∞) > 0 LN ∼ c · N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 48

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Case TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) ≡ ∂

∂δφ(δ, ∞) ≡ 0 LN = o(N)

◮ If δ > 0 then

∂ ∂δφ(δ, ∞) > 0 LN ∼ c · 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 16, 2010

SLIDE 49

Introduction The Model Free Energy Path Behavior The Proof

The free energy: sumup

Case TN ≡ T < ∞

◮ φ(δ, T) is analytic on R: no phase transitions ◮ ∂ ∂δφ(δ, T) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c · N ◮ Path behavior: diffusive scaling of SN

Case TN → ∞

◮ Phase transition (only) at δ = 0

◮ If δ ≤ 0 then φ(δ, ∞) ≡ ∂

∂δφ(δ, ∞) ≡ 0 LN = o(N)

◮ If δ > 0 then

∂ ∂δφ(δ, ∞) > 0 LN ∼ c · 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 16, 2010

SLIDE 50

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 51

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 52

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞? For δ < 0, φ(δ, T) = − π2 2T 2 + cδ T 3 + o 1 T 3

with cδ > 0 explicit.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 53

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞? For δ < 0, φ(δ, T) = − π2 2T 2 + cδ T 3 + o 1 T 3

with cδ > 0 explicit.

[Owczarek et al. 2008] prove (for the polymer in a slit) that Z T

N,δ ≈ exp

− π2

2T 2 N + o N T 2

as N, T → ∞ .

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 54

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞? For δ < 0, φ(δ, T) = − π2 2T 2 + cδ T 3 + o 1 T 3

with cδ > 0 explicit.

[Owczarek et al. 2008] prove (for the polymer in a slit) that Z T

N,δ ≈ exp

− π2

2T 2 N + o N T 2

as N, T → ∞ .

We show that φ(δ, T) can be developed at wish in (⋆).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 55

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞? For δ < 0, φ(δ, T) = − π2 2T 2 + cδ T 3 + o 1 T 3

with cδ > 0 explicit.

[Owczarek et al. 2008] prove (for the polymer in a slit) that Z T

N,δ ≈ exp

− π2

2T 2 N + o N T 2

as N, T → ∞ .

We show that φ(δ, T) can be developed at wish in (⋆).

◮ Force exerted by the polymer on the confining walls (slit)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 56

Introduction The Model Free Energy Path Behavior The Proof

Beyond the free energy

Free energy says that for fixed T Z T

N,δ ≈ exp

φ(δ, T) · N
as N → ∞ .

(⋆) What if both N, T → ∞? For δ < 0, φ(δ, T) = − π2 2T 2 + cδ T 3 + o 1 T 3

with cδ > 0 explicit.

[Owczarek et al. 2008] prove (for the polymer in a slit) that Z T

N,δ ≈ exp

− π2

2T 2 N + o N T 2

as N, T → ∞ .

We show that φ(δ, T) can be developed at wish in (⋆).

◮ Force exerted by the polymer on the confining walls (slit) ◮ Path behavior (multi-interface)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 57

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 58

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Henceforth TN → ∞.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 59

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Henceforth TN → ∞. For δ > 0 positive density of contacts with the interfaces: LN ∼ Cδ · N with Cδ = ∂φ

∂δ (δ, ∞) > 0.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 60

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Henceforth TN → ∞. For δ > 0 positive density of contacts with the interfaces: LN ∼ Cδ · N with Cδ = ∂φ

∂δ (δ, ∞) > 0.

Simple homogeneous pinning model (one interface at zero):

N max |Si| ≈ log N SN = O(1)

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 61

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Henceforth TN → ∞. For δ > 0 positive density of contacts with the interfaces: LN ∼ Cδ · N with Cδ = ∂φ

∂δ (δ, ∞) > 0.

Simple homogeneous pinning model (one interface at zero):

N TN max |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 16, 2010

SLIDE 62

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Henceforth TN → ∞. For δ > 0 positive density of contacts with the interfaces: LN ∼ Cδ · N with Cδ = ∂φ

∂δ (δ, ∞) > 0.

Simple homogeneous pinning model (one interface at zero):

N TN

◮ If TN ≫ log N nothing changes: polymer localized at zero ◮ If TN ≪ log N different interfaces worth visiting

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 63

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

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 16, 2010

SLIDE 64

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Under PTN

N,δ (N ≫ 1), how long is the time ˆ

τ T

1 to reach level ±T?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 65

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Under PTN

N,δ (N ≫ 1), how long is the time ˆ

τ T

1 to reach level ±T?

It turns out that ˆ τ T

1 ≈ ecδT with cδ > 0.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 66

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Under PTN

N,δ (N ≫ 1), how long is the time ˆ

τ T

1 to reach level ±T?

It turns out that ˆ τ T

1 ≈ ecδT with cδ > 0. ◮ If ecδTN ≪ N, there are ≈ N/ecδTN jumps to a neighboring

interface, therefore SN ≍ TN

N

ecδTN = (e− cδ

2 TNTN)

√ N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 67

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Under PTN

N,δ (N ≫ 1), how long is the time ˆ

τ T

1 to reach level ±T?

It turns out that ˆ τ T

1 ≈ ecδT with cδ > 0. ◮ If ecδTN ≪ N, there are ≈ N/ecδTN jumps to a neighboring

interface, therefore SN ≍ TN

N

ecδTN = (e− cδ

2 TNTN)

√ N

◮ If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,

hence SN ≈ TN.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 68

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: heuristics

Notation

SN ≍ αN means SN/αN is tight and PTN

N,δ(|SN/αN| ≥ ε) ≥ ε

Under PTN

N,δ (N ≫ 1), how long is the time ˆ

τ T

1 to reach level ±T?

It turns out that ˆ τ T

1 ≈ ecδT with cδ > 0. ◮ If ecδTN ≪ N, there are ≈ N/ecδTN jumps to a neighboring

interface, therefore SN ≍ TN

N

ecδTN = (e− cδ

2 TNTN)

√ N

◮ If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,

hence SN ≈ TN.

◮ If ecδTN ≫ N, there are no jumps to a neighboring interface,

hence SN ≍ O(1).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 69

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 70

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1) Note that TN ≪ SN ≪ √ N.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 71

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1) 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 16, 2010

SLIDE 72

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1) 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 16, 2010

SLIDE 73

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1) Note that TN ≪ SN ≪ √ N.

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

SN TN = ⇒ SΓ Γ ∼ Poisson(f (δ, ζ))

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

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 74

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 SN Cδ (e− cδ

2 TNTN)

√ N = ⇒ N(0, 1) Note that TN ≪ SN ≪ √ N.

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

SN TN = ⇒ SΓ Γ ∼ Poisson(f (δ, ζ))

◮ 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 16, 2010

SLIDE 75

Introduction The Model Free Energy Path Behavior The Proof

The attractive case δ > 0: path results

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 16, 2010

SLIDE 76

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

Again TN → ∞.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 77

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

Again TN → ∞. For δ < 0, zero density of contacts with the interfaces: LN = o(N).

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 78

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

Again TN → ∞. For δ < 0, zero density of contacts with the interfaces: LN = o(N). Simple homogeneous pinning model (one interface at zero):

N S N = O ( N )

1/2 y x

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 79

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

Again TN → ∞. For δ < 0, zero density of contacts with the interfaces: LN = o(N). Simple homogeneous pinning model (one interface at zero):

N S N = O ( N )

1/2

T N

x y

◮ If TN ≫

√ N nothing changes: SN ≍ √ N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 80

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

Again TN → ∞. For δ < 0, zero density of contacts with the interfaces: LN = o(N). Simple homogeneous pinning model (one interface at zero):

N S N = ?

y x

T N

◮ If TN ≫

√ N nothing changes: SN ≍ √ N

◮ If TN ≪

√ N does the polymer visit other interfaces?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 81

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

How long is the time ˆ τ T

1 to reach level ±T?

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 82

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

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 16, 2010

SLIDE 83

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

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 16, 2010

SLIDE 84

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

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 TN ≪ N1/3, there are ≈ N/T 3 N ≫ 1 jumps to a neighboring

interface, hence SN ≈ TN

N/T 3

N =

N/TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 85

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: heuristics

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 TN ≪ N1/3, there are ≈ N/T 3 N ≫ 1 jumps to a neighboring

interface, hence SN ≈ TN

N/T 3

N =

N/TN

◮ if N1/3 ≪ TN ≪

√ N, there are no jumps to a neighboring interface confinement: SN ≍ TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 86

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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 16, 2010

SLIDE 87

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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 16, 2010

SLIDE 88

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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)

◮ If TN ∼ (const.)N1/3 then SN ≍ TN

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 89

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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)

◮ If TN ∼ (const.)N1/3 then SN ≍ TN ◮ If (const.)N1/3 ≤ TN ≤ (const.)

√ N then SN ≍ TN and ∀ǫ∃L : PTN

N,δ

0 < |Sn| < TN, ∀n ∈ {L, . . . , N}
≥ 1 − ǫ

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 90

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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)

◮ If TN ∼ (const.)N1/3 then SN ≍ TN ◮ If (const.)N1/3 ≤ TN ≤ (const.)

√ N then SN ≍ TN and ∀ǫ∃L : PTN

N,δ

0 < |Sn| < TN, ∀n ∈ {L, . . . , N}
≥ 1 − ǫ

◮ If TN ≫

√ N then SN ≍ √ N

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 91

Introduction The Model Free Energy Path Behavior The Proof

The repulsive case δ < 0: path results

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 16, 2010

SLIDE 92

Introduction The Model Free Energy Path Behavior The Proof

Outline

1. Introduction and motivations
2. Definition of the model
3. The free energy
4. Path results
5. Techniques from the proof

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 93

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 94

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 95

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 96

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 97

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 98

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 99

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 100

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 101

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 102

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 103

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 104

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 105

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 106

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 107

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 108

Introduction The Model Free Energy Path Behavior The Proof

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 16, 2010

SLIDE 109

Introduction The Model Free Energy Path Behavior The Proof

Thanks.

Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

SLIDE 110

Introduction The Model Free Energy Path Behavior 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 16, 2010