a polymer in a multi-interface...
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Introduction The Model Free Energy Path Behavior The Proof
A Polymer in a Multi-Interface Medium
Francesco Caravenna
Universita degli Studi di Milano-Bicocca
Joint work with Nicolas Petrelis (Nantes)
Universitat Bonn ∼ December 16, 2010
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 2: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/2.jpg)
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
![Page 3: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/3.jpg)
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
![Page 4: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/4.jpg)
Introduction The Model Free Energy Path Behavior The Proof
A polymer in a multi-interface medium
Single interface case is well understood.
Copolymer interaction [den Hollander & Wuthrich JSP 04]Some path results for log logN�TN� logN (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
![Page 5: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/5.jpg)
Introduction The Model Free Energy Path Behavior The Proof
A polymer in a multi-interface medium
Single interface case is well understood.
Copolymer interaction [den Hollander & Wuthrich JSP 04]Some path results for log logN�TN� logN (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
![Page 6: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/6.jpg)
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 & Wuthrich JSP 04]Some path results for log logN�TN� logN (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
![Page 7: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/7.jpg)
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 & Wuthrich JSP 04]Some path results for log logN�TN� logN (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
![Page 8: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/8.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Stabilization of colloidal dispersions
The addition of polymers into a colloid can prevent the aggregationof droplets via entropic repulsion (steric stabilization of the colloid)
Polymer confined between two walls and interacting with them:(polymer in a slit)
0
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
![Page 9: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/9.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Stabilization of colloidal dispersions
The addition of polymers into a colloid can prevent the aggregationof droplets via entropic repulsion (steric stabilization of the colloid)
Polymer confined between two walls and interacting with them:(polymer in a slit)
0
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
![Page 10: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/10.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Stabilization of colloidal dispersions
The addition of polymers into a colloid can prevent the aggregationof droplets via entropic repulsion (steric stabilization of the colloid)
Polymer confined between two walls and interacting with them:(polymer in a slit)
0
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
![Page 11: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/11.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Multi-interface medium vs. slit
0
0
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
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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
![Page 13: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/13.jpg)
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 PTN,δ on paths
I (1 + 1)-dimensionale model: {(i ,Si )}i≥0
I PTN,δ absolutely continuous w.r.t. SRW {Si}i≥0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 14: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/14.jpg)
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 PTN,δ on paths
I (1 + 1)-dimensionale model: {(i ,Si )}i≥0
I PTN,δ absolutely continuous w.r.t. SRW {Si}i≥0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 15: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/15.jpg)
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 PTN,δ on paths
I (1 + 1)-dimensionale model: {(i ,Si )}i≥0
I PTN,δ absolutely continuous w.r.t. SRW {Si}i≥0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 16: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/16.jpg)
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 PTN,δ on paths
I (1 + 1)-dimensionale model: {(i , Si )}i≥0
I PTN,δ absolutely continuous w.r.t. SRW {Si}i≥0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 17: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/17.jpg)
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 PTN,δ on paths
I (1 + 1)-dimensionale model: {(i , Si )}i≥0
I PTN,δ absolutely continuous w.r.t. SRW {Si}i≥0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 18: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/18.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)
=1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 19: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/19.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)
=1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 20: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/20.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)
=1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 21: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/21.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)
=1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 22: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/22.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)
=1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 23: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/23.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Ingredients of PTN,δ:
I 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) = 12 .
I Polymer size N ∈ 2N (number of monomers)
I Interaction strength δ ∈ R (> 0 attractive, < 0 repulsive)
I Interface distance T ∈ 2N (interfaces ≡ TZ)
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)=
1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 24: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/24.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Definition
Sn
N
T
Polymer measure:
dPTN,δ
dP(S) :=
1
ZTN,δ
exp(δLN(S)
)=
1
ZTN,δ
exp
{δ
N∑i=1
1{Si ∈TZ}
}
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 25: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/25.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 26: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/26.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 27: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/27.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 28: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/28.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 29: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/29.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 30: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/30.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 31: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/31.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Some questions
The law PTN,δ describes the statistical distribution of the
configurations of the polymer, given the external conditions
We are interested in the properties of PTN,δ as N →∞
(thermodynamic limit), for fixed δ ∈ R and for arbitrary T = TN
Some questions on PTNN,δ for large N
I What is the typical size of SN?
I How many monomers touch the interfaces?
I How many different interfaces are visited?
I 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
![Page 32: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/32.jpg)
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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy
The free energy φ(δ, {Tn}n) encodes the exponential asymptoticbehavior of the normalization constant ZTN
N,δ (partition function)
φ(δ, {Tn}n) := limN→∞
1
Nlog ZTN
N,δ (super-additivity + . . . )
= limN→∞
1
Nlog 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
(LNN
) 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy
The free energy φ(δ, {Tn}n) encodes the exponential asymptoticbehavior of the normalization constant ZTN
N,δ (partition function)
φ(δ, {Tn}n) := limN→∞
1
Nlog ZTN
N,δ (super-additivity + . . . )
= limN→∞
1
Nlog 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
(LNN
) 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy
The free energy φ(δ, {Tn}n) encodes the exponential asymptoticbehavior of the normalization constant ZTN
N,δ (partition function)
φ(δ, {Tn}n) := limN→∞
1
Nlog ZTN
N,δ (super-additivity + . . . )
= limN→∞
1
Nlog 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
(LNN
) 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy
The free energy φ(δ, {Tn}n) encodes the exponential asymptoticbehavior of the normalization constant ZTN
N,δ (partition function)
φ(δ, {Tn}n) := limN→∞
1
Nlog ZTN
N,δ (super-additivity + . . . )
= limN→∞
1
Nlog 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
(LNN
) 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: characterization
We assume that TN → T ∈ 2N ∪ {∞}, i.e.,
I either TN ≡ T <∞ is constant for N large
I or TN →∞ as N →∞ (T =∞)
Let τT1 := inf{n > 0 : Sn ∈ {±T , 0}
}and set
QT (λ) := E(e−λτ
T1)
I if T <∞, QT : (−λT ,∞)→ (0,∞) (λT ∼ − π2
2T 2 )
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: characterization
We assume that TN → T ∈ 2N ∪ {∞}, i.e.,
I either TN ≡ T <∞ is constant for N large
I or TN →∞ as N →∞ (T =∞)
Let τT1 := inf{n > 0 : Sn ∈ {±T , 0}
}and set
QT (λ) := E(e−λτ
T1)
I if T <∞, QT : (−λT ,∞)→ (0,∞) (λT ∼ − π2
2T 2 )
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: characterization
We assume that TN → T ∈ 2N ∪ {∞}, i.e.,
I either TN ≡ T <∞ is constant for N large
I or TN →∞ as N →∞ (T =∞)
Let τT1 := inf{n > 0 : Sn ∈ {±T , 0}
}and set
QT (λ) := E(e−λτ
T1)
I if T <∞, QT : (−λT ,∞)→ (0,∞) (λT ∼ − π2
2T 2 )
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: characterization
We assume that TN → T ∈ 2N ∪ {∞}, i.e.,
I either TN ≡ T <∞ is constant for N large
I or TN →∞ as N →∞ (T =∞)
Let τT1 := inf{n > 0 : Sn ∈ {±T , 0}
}and set
QT (λ) := E(e−λτ
T1)
I if T <∞, QT : (−λT ,∞)→ (0,∞) (λT ∼ − π2
2T 2 )
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: characterization
We assume that TN → T ∈ 2N ∪ {∞}, i.e.,
I either TN ≡ T <∞ is constant for N large
I or TN →∞ as N →∞ (T =∞)
Let τT1 := inf{n > 0 : Sn ∈ {±T , 0}
}and set
QT (λ) := E(e−λτ
T1)
I if T <∞, QT : (−λT ,∞)→ (0,∞) (λT ∼ − π2
2T 2 )
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The free energy: sumup
Case TN ≡ T <∞I φ(δ,T ) is analytic on R: no phase transitions
I ∂∂δφ(δ,T ) > 0 ∀δ ∈ R: positive density of contacts LN ∼ c ·N
I Path behavior: diffusive scaling of SN
Case TN →∞I Phase transition (only) at δ = 0
I If δ ≤ 0 then φ(δ,∞) ≡ ∂∂δφ(δ,∞) ≡ 0 LN = o(N)
I If δ > 0 then ∂∂δφ(δ,∞) > 0 LN ∼ c · N
I Every {TN}N →∞ yields the same free energy as if TN ≡ ∞(homogeneous pinning model) same density of visits
I Same path behavior? NO!
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Beyond the free energy
Free energy says that for fixed T
ZTN,δ ≈ exp
(φ(δ,T ) · N
)as N →∞ . (?)
What if both N,T →∞?
For δ < 0, φ(δ,T ) = − π2
2T 2+
cδT 3
+ o
(1
T 3
)with cδ > 0 explicit.
[Owczarek et al. 2008] prove (for the polymer in a slit) that
ZTN,δ ≈ exp
(− π2
2T 2N + o
(N
T 2
))as N,T →∞ .
We show that φ(δ,T ) can be developed at wish in (?).
I Force exerted by the polymer on the confining walls (slit)
I Path behavior (multi-interface)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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
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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):
I If TN � logN nothing changes: polymer localized at zero
I If TN � logN different interfaces worth visiting
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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):
I If TN � logN nothing changes: polymer localized at zero
I If TN � logN different interfaces worth visiting
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 NSN = O(1)
I If TN � logN nothing changes: polymer localized at zero
I If TN � logN different interfaces worth visiting
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 NSN = O(1)
I If TN � logN nothing changes: polymer localized at zero
I If TN � logN different interfaces worth visiting
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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
I If TN � logN nothing changes: polymer localized at zero
I If TN � logN different interfaces worth visiting
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: heuristics
Notation
SN � αN means SN/αN is tight and PTNN,δ(|SN/αN | ≥ ε) ≥ ε
Under PTNN,δ (N � 1), how long is the time τT1 to reach level ±T?
It turns out that τT1 ≈ ecδT with cδ > 0.
I If ecδTN � N, there are ≈ N/ecδTN jumps to a neighboringinterface, therefore
SN � TN
√N
ecδTN= (e−
cδ2TNTN)
√N
I If ecδTN ≈ N, there are O(1) jumps to a neighboring interface,hence SN ≈ TN .
I 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
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN − 1cδ
logN → −∞ then SN � (e−cδ2TN TN)
√N
SN
Cδ (e−cδ2TNTN)
√N
=⇒ N(0, 1)
Note that TN � SN �√N.
I If TN − 1cδ
logN → ζ ∈ R then SN � TN
SNTN
=⇒ SΓ Γ ∼ Poisson(f (δ, ζ))
I If TN − 1cδ
logN → +∞ then SN � O(1)
limL→∞
supN∈2N
PTNN,δ
(|SN | > L
)= 0
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The attractive case δ > 0: path results
SN√N
1cδlogN
1cδlogN
O(1)
1 TN
• Sub-diffusive scaling (TN →∞) • Transition at TN ≈ logN
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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):
I If TN �√N nothing changes: SN �
√N
I If TN �√N does the polymer visit other interfaces?
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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):
I If TN �√N nothing changes: SN �
√N
I If TN �√N does the polymer visit other interfaces?
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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
I If TN �√N nothing changes: SN �
√N
I If TN �√N does the polymer visit other interfaces?
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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
I If TN �√N nothing changes: SN �
√N
I If TN �√N does the polymer visit other interfaces?
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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
I If TN �√N nothing changes: SN �
√N
I If TN �√N does the polymer visit other interfaces?
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: heuristics
How long is the time τT1 to reach level ±T?
Under the simple random walk law τT1 ≈ T 2 (diffusivity)
Under the polymer measure PTN,δ with δ < 0 τT1 ≈ T 3 (repulsion)
I if TN � N1/3, there are ≈ N/T 3N � 1 jumps to a neighboring
interface, hence SN ≈ TN
√N/T 3
N =√N/TN
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: heuristics
How long is the time τT1 to reach level ±T?
N
T
τT1 ≈ T2
Under the simple random walk law τT1 ≈ T 2 (diffusivity)
Under the polymer measure PTN,δ with δ < 0 τT1 ≈ T 3 (repulsion)
I if TN � N1/3, there are ≈ N/T 3N � 1 jumps to a neighboring
interface, hence SN ≈ TN
√N/T 3
N =√N/TN
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: heuristics
How long is the time τT1 to reach level ±T?
N
T
τT1 ≈ T3
Under the simple random walk law τT1 ≈ T 2 (diffusivity)
Under the polymer measure PTN,δ with δ < 0 τT1 ≈ T 3 (repulsion)
I if TN � N1/3, there are ≈ N/T 3N � 1 jumps to a neighboring
interface, hence SN ≈ TN
√N/T 3
N =√N/TN
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: heuristics
How long is the time τT1 to reach level ±T?
N
T
τT1 ≈ T3
Under the simple random walk law τT1 ≈ T 2 (diffusivity)
Under the polymer measure PTN,δ with δ < 0 τT1 ≈ T 3 (repulsion)
I if TN � N1/3, there are ≈ N/T 3N � 1 jumps to a neighboring
interface, hence SN ≈ TN
√N/T 3
N =√N/TN
I 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
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: heuristics
How long is the time τT1 to reach level ±T?
N
T
τT1 ≈ T3
Under the simple random walk law τT1 ≈ T 2 (diffusivity)
Under the polymer measure PTN,δ with δ < 0 τT1 ≈ T 3 (repulsion)
I if TN � N1/3, there are ≈ N/T 3N � 1 jumps to a neighboring
interface, hence SN ≈ TN
√N/T 3
N =√N/TN
I 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
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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 PTNN,δ:
I If TN � N1/3 then SN �√N/TN � TN
c1 P(a < Z ≤ b
)≤ PTN
N,δ
(a <
SN
Cδ
√NTN
≤ b
)≤ c2 P
(a < Z ≤ b
)with Z ∼ N(0, 1)
I If TN ∼ (const.)N1/3 then SN � TN
I If (const.)N1/3 ≤ TN ≤ (const.)√N then SN � TN and
∀ε∃L : PTNN,δ
(0 < |Sn| < TN , ∀n ∈ {L, . . . ,N}
)≥ 1− ε
I If TN �√N then SN �
√N
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN � N1/3 then SN �√N/TN � TN
c1 P(a < Z ≤ b
)≤ PTN
N,δ
(a <
SN
Cδ
√NTN
≤ b
)≤ c2 P
(a < Z ≤ b
)with Z ∼ N(0, 1)
I If TN ∼ (const.)N1/3 then SN � TN
I If (const.)N1/3 ≤ TN ≤ (const.)√N then SN � TN and
∀ε∃L : PTNN,δ
(0 < |Sn| < TN , ∀n ∈ {L, . . . ,N}
)≥ 1− ε
I If TN �√N then SN �
√N
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN � N1/3 then SN �√N/TN � TN
c1 P(a < Z ≤ b
)≤ PTN
N,δ
(a <
SN
Cδ
√NTN
≤ b
)≤ c2 P
(a < Z ≤ b
)with Z ∼ N(0, 1)
I If TN ∼ (const.)N1/3 then SN � TN
I If (const.)N1/3 ≤ TN ≤ (const.)√N then SN � TN and
∀ε∃L : PTNN,δ
(0 < |Sn| < TN , ∀n ∈ {L, . . . ,N}
)≥ 1− ε
I If TN �√N then SN �
√N
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN � N1/3 then SN �√N/TN � TN
c1 P(a < Z ≤ b
)≤ PTN
N,δ
(a <
SN
Cδ
√NTN
≤ b
)≤ c2 P
(a < Z ≤ b
)with Z ∼ N(0, 1)
I If TN ∼ (const.)N1/3 then SN � TN
I If (const.)N1/3 ≤ TN ≤ (const.)√N then SN � TN and
∀ε∃L : PTNN,δ
(0 < |Sn| < TN , ∀n ∈ {L, . . . ,N}
)≥ 1− ε
I If TN �√N then SN �
√N
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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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 PTNN,δ:
I If TN � N1/3 then SN �√N/TN � TN
c1 P(a < Z ≤ b
)≤ PTN
N,δ
(a <
SN
Cδ
√NTN
≤ b
)≤ c2 P
(a < Z ≤ b
)with Z ∼ N(0, 1)
I If TN ∼ (const.)N1/3 then SN � TN
I If (const.)N1/3 ≤ TN ≤ (const.)√N then SN � TN and
∀ε∃L : PTNN,δ
(0 < |Sn| < TN , ∀n ∈ {L, . . . ,N}
)≥ 1− ε
I If TN �√N then SN �
√N
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
The repulsive case δ < 0: path results
SN
√N
√NN1/3
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
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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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Let τT1 , τT2 , τ
T3 . . . be the points at which Sn visits an interface
τTk+1 := inf{n > τTk : Sn − SτTk
∈ {−T , 0,T}}
(T is fixed)
Under the simple random walk law {τTn }n∈N is a classical renewalprocess with explicit law
qT (n) := P(τT1 = n) ≈ 1
(min{n,T 2})3/2e−
π2
2T2 n
Note that under P
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Let τT1 , τT2 , τ
T3 . . . be the points at which Sn visits an interface
τTk+1 := inf{n > τTk : Sn − SτTk
∈ {−T , 0,T}}
(T is fixed)
Under the simple random walk law {τTn }n∈N is a classical renewalprocess with explicit law
qT (n) := P(τT1 = n) ≈ 1
(min{n,T 2})3/2e−
π2
2T2 n
Note that under P
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Let τT1 , τT2 , τ
T3 . . . be the points at which Sn visits an interface
τTk+1 := inf{n > τTk : Sn − SτTk
∈ {−T , 0,T}}
(T is fixed)
Under the simple random walk law {τTn }n∈N is a classical renewalprocess with explicit law
qT (n) := P(τT1 = n) ≈ 1
(min{n,T 2})3/2e−
π2
2T2 n
Note that under P
τT1 ≈
O(1) with probab. 1− 1
T
[qT (n) ≈ 1
n3/2
]
O(T 2) with probab. 1T
[qT (n) ≈ 1
T 3 e− π2
2T2 n]
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Let τT1 , τT2 , τ
T3 . . . be the points at which Sn visits an interface
τTk+1 := inf{n > τTk : Sn − SτTk
∈ {−T , 0,T}}
(T is fixed)
Under the simple random walk law {τTn }n∈N is a classical renewalprocess with explicit law
qT (n) := P(τT1 = n) ≈ 1
(min{n,T 2})3/2e−
π2
2T2 n
Note that under P
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Under the polymer measure PTN,δ the process {τTn }n∈N is not even
time-homogeneous . . .
however for large N it is nearly a renewalprocess with a different law Pδ,T : for both δ > 0 and δ < 0
Kδ,T (n) := Pδ,T (τT1 = n) = eδ P(τT1 = 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 + CδT 3 as T →∞, hence
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Under the polymer measure PTN,δ the process {τTn }n∈N is not even
time-homogeneous . . . however for large N it is nearly a renewalprocess with a different law Pδ,T : for both δ > 0 and δ < 0
Kδ,T (n) := Pδ,T (τT1 = n) = eδ P(τT1 = 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 + CδT 3 as T →∞, hence
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Under the polymer measure PTN,δ the process {τTn }n∈N is not even
time-homogeneous . . . however for large N it is nearly a renewalprocess with a different law Pδ,T : for both δ > 0 and δ < 0
Kδ,T (n) := Pδ,T (τT1 = n) = eδ P(τT1 = 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 + CδT 3 as T →∞, hence
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Under the polymer measure PTN,δ the process {τTn }n∈N is not even
time-homogeneous . . . however for large N it is nearly a renewalprocess with a different law Pδ,T : for both δ > 0 and δ < 0
Kδ,T (n) := Pδ,T (τT1 = n) = eδ P(τT1 = 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 + CδT 3 as T →∞, hence
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
A renewal theory approach
Under the polymer measure PTN,δ the process {τTn }n∈N is not even
time-homogeneous . . . however for large N it is nearly a renewalprocess with a different law Pδ,T : for both δ > 0 and δ < 0
Kδ,T (n) := Pδ,T (τT1 = n) = eδ P(τT1 = 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 + CδT 3 as T →∞, hence
τT1 ≈
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
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Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
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Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 106: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/106.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 107: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/107.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 108: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/108.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Strategy of the proof
For fixed T , the law of τT ∩ [0,N] = {τT1 , . . . , τTLN} is the same
under PTN,δ
(·∣∣N ∈ τT ) and Pδ,T
(·∣∣N ∈ τT )
• Pδ,T does not depend explicitly on N • PTN,δ =⇒ Pδ,T
However we want to study PTNN,δ with TN varying with N
1. Study τTN ∩ [0,N] under Pδ,TN
2. Transfer the results to Pδ,TN( · |N ∈ τTN ) (hard part)
In this way we control PTNN,δ( · |N ∈ τTN )
3. Remove the conditioning on {N ∈ τTN}
I Good estimates on qT (n) and on the free energy φ(δ,T )
I Uniform renewal theorems
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
![Page 109: A Polymer in a Multi-Interface Mediumstaff.matapp.unimib.it/~fcaraven/download/slides/bonn_16-12-2010.… · IntroductionThe ModelFree EnergyPath BehaviorThe Proof A Polymer in a](https://reader034.vdocuments.us/reader034/viewer/2022050508/5f997f1d73cf3179354665b0/html5/thumbnails/109.jpg)
Introduction The Model Free Energy Path Behavior The Proof
Thanks.
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010
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Introduction The Model Free Energy Path Behavior The Proof
References
I [CP1] F. Caravenna and N. PetrelisA polymer in a multi-interface mediumAAP (2009)
I [CP2] F. Caravenna and N. PetrelisDepinning of a polymer in a multi-interface mediumEJP (2009)
Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010