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

Introduction The Model Free Energy Path Behavior The Proof Some questions The law P T N ,δ describes the statistical distribution of the configurations of the polymer, given the external conditions We are interested in the properties of P T N ,δ as N → ∞ (thermodynamic limit), for fixed δ ∈ R and for arbitrary T = T N Some questions on P T N N ,δ for large N ◮ What is the typical size of S N ? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof Some questions The law P T N ,δ describes the statistical distribution of the configurations of the polymer, given the external conditions We are interested in the properties of P T N ,δ as N → ∞ (thermodynamic limit), for fixed δ ∈ R and for arbitrary T = T N Some questions on P T N N ,δ for large N ◮ What is the typical size of S N ? ◮ How many monomers touch the interfaces? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof Some questions The law P T N ,δ describes the statistical distribution of the configurations of the polymer, given the external conditions We are interested in the properties of P T N ,δ as N → ∞ (thermodynamic limit), for fixed δ ∈ R and for arbitrary T = T N Some questions on P T N N ,δ for large N ◮ What is the typical size of S N ? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof Some questions The law P T N ,δ describes the statistical distribution of the configurations of the polymer, given the external conditions We are interested in the properties of P T N ,δ as N → ∞ (thermodynamic limit), for fixed δ ∈ R and for arbitrary T = T N Some questions on P T N N ,δ for large N ◮ What is the typical size of S N ? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited? ◮ Interplay between the parameters δ and T = { T N } N ? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof Some questions The law P T N ,δ describes the statistical distribution of the configurations of the polymer, given the external conditions We are interested in the properties of P T N ,δ as N → ∞ (thermodynamic limit), for fixed δ ∈ R and for arbitrary T = T N Some questions on P T N N ,δ for large N ◮ What is the typical size of S N ? ◮ How many monomers touch the interfaces? ◮ How many different interfaces are visited? ◮ Interplay between the parameters δ and T = { T N } N ? Penalization of the simple random walk Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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

Introduction The Model Free Energy Path Behavior The Proof The free energy The free energy φ ( δ, { T n } n ) encodes the exponential asymptotic behavior of the normalization constant Z T N N ,δ (partition function) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy The free energy φ ( δ, { T n } n ) encodes the exponential asymptotic behavior of the normalization constant Z T N N ,δ (partition function) 1 N log Z T N φ ( δ, { T n } n ) := lim (super-additivity + . . . ) N ,δ N →∞ 1 � � �� = lim N log E exp δ L N N →∞ L N := # { i ≤ N : S i ∈ T Z } number of visits to the interfaces Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy The free energy φ ( δ, { T n } n ) encodes the exponential asymptotic behavior of the normalization constant Z T N N ,δ (partition function) 1 N log Z T N φ ( δ, { T n } n ) := lim (super-additivity + . . . ) N ,δ N →∞ 1 � � �� = lim N log E exp δ L N N →∞ L N := # { i ≤ N : S i ∈ T Z } number of visits to the interfaces φ is a generating function: ∂ � L N � N →∞ E T N L N ∼ ∂φ ∂δφ ( δ, { T n } n ) = lim ∂δ · N � N ,δ N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy The free energy φ ( δ, { T n } n ) encodes the exponential asymptotic behavior of the normalization constant Z T N N ,δ (partition function) 1 N log Z T N φ ( δ, { T n } n ) := lim (super-additivity + . . . ) N ,δ N →∞ 1 � � �� = lim N log E exp δ L N N →∞ L N := # { i ≤ N : S i ∈ T Z } number of visits to the interfaces φ is a generating function: ∂ � L N � N →∞ E T N L N ∼ ∂φ ∂δφ ( δ, { T n } n ) = lim ∂δ · N � N ,δ N φ ( δ, { T n } n ) non-analytic at δ ← → phase transition at δ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: characterization We assume that T N → T ∈ 2 N ∪ {∞} , i.e., ◮ either T N ≡ T < ∞ is constant for N large ◮ or T N → ∞ as N → ∞ ( T = ∞ ) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: characterization We assume that T N → T ∈ 2 N ∪ {∞} , i.e., ◮ either T N ≡ T < ∞ is constant for N large ◮ or T N → ∞ as N → ∞ ( T = ∞ ) Let τ T � � 1 := inf n > 0 : S n ∈ {± T , 0 } and set e − λτ T � 1 � Q T ( λ ) := E Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: characterization We assume that T N → T ∈ 2 N ∪ {∞} , i.e., ◮ either T N ≡ T < ∞ is constant for N large ◮ or T N → ∞ as N → ∞ ( T = ∞ ) Let τ T � � 1 := inf n > 0 : S n ∈ {± T , 0 } and set e − λτ T � 1 � Q T ( λ ) := E ( λ T ∼ − π 2 ◮ if T < ∞ , Q T : ( − λ T , ∞ ) → (0 , ∞ ) 2 T 2 ) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: characterization We assume that T N → T ∈ 2 N ∪ {∞} , i.e., ◮ either T N ≡ T < ∞ is constant for N large ◮ or T N → ∞ as N → ∞ ( T = ∞ ) Let τ T � � 1 := inf n > 0 : S n ∈ {± T , 0 } and set e − λτ T � 1 � Q T ( λ ) := E ( λ T ∼ − π 2 ◮ if T < ∞ , Q T : ( − λ T , ∞ ) → (0 , ∞ ) 2 T 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

Introduction The Model Free Energy Path Behavior The Proof The free energy: characterization We assume that T N → T ∈ 2 N ∪ {∞} , i.e., ◮ either T N ≡ T < ∞ is constant for N large ◮ or T N → ∞ as N → ∞ ( T = ∞ ) Let τ T � � 1 := inf n > 0 : S n ∈ {± T , 0 } and set e − λτ T � 1 � Q T ( λ ) := E ( λ T ∼ − π 2 ◮ if T < ∞ , Q T : ( − λ T , ∞ ) → (0 , ∞ ) 2 T 2 ) ◮ if T = ∞ , Q ∞ : [0 , ∞ ) → (0 , 1] (first return to zero of SRW) Theorem ([CP1]). Let T N → T . � − 1 ( e − δ ) �� Q T if T < + ∞ φ ( δ, { T n } n ) = φ ( δ, T ) = � − 1 ( e − δ ∧ 1) � Q ∞ if T = + ∞ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Case T N → ∞ ◮ Phase transition (only) at δ = 0 Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Case T N → ∞ ◮ Phase transition (only) at δ = 0 ◮ If δ ≤ 0 then φ ( δ, ∞ ) ≡ ∂ ∂δ φ ( δ, ∞ ) ≡ 0 � L N = o ( N ) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Case T N → ∞ ◮ Phase transition (only) at δ = 0 ◮ If δ ≤ 0 then φ ( δ, ∞ ) ≡ ∂ ∂δ φ ( δ, ∞ ) ≡ 0 � L N = o ( N ) ∂ ◮ If δ > 0 then ∂δ φ ( δ, ∞ ) > 0 � L N ∼ c · N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Case T N → ∞ ◮ Phase transition (only) at δ = 0 ◮ If δ ≤ 0 then φ ( δ, ∞ ) ≡ ∂ ∂δ φ ( δ, ∞ ) ≡ 0 � L N = o ( N ) ∂ ◮ If δ > 0 then ∂δ φ ( δ, ∞ ) > 0 � L N ∼ c · N ◮ Every { T N } N → ∞ yields the same free energy as if T N ≡ ∞ (homogeneous pinning model) � same density of visits Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The free energy: sumup Case T N ≡ T < ∞ ◮ φ ( δ, T ) is analytic on R : no phase transitions ∂ ∂δ φ ( δ, T ) > 0 ∀ δ ∈ R : positive density of contacts L N ∼ c · N ◮ ◮ Path behavior: diffusive scaling of S N Case T N → ∞ ◮ Phase transition (only) at δ = 0 ◮ If δ ≤ 0 then φ ( δ, ∞ ) ≡ ∂ ∂δ φ ( δ, ∞ ) ≡ 0 � L N = o ( N ) ∂ ◮ If δ > 0 then ∂δ φ ( δ, ∞ ) > 0 � L N ∼ c · N ◮ Every { T N } N → ∞ yields the same free energy as if T N ≡ ∞ (homogeneous pinning model) � same density of visits ◮ Same path behavior? NO! Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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

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

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 → ∞ ? � 1 For δ < 0, φ ( δ, T ) = − π 2 � 2 T 2 + c δ T 3 + o with c δ > 0 explicit. T 3 Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 → ∞ ? � 1 For δ < 0, φ ( δ, T ) = − π 2 � 2 T 2 + c δ T 3 + o with c δ > 0 explicit. T 3 [Owczarek et al. 2008] prove (for the polymer in a slit) that � N − π 2 � �� Z T N ,δ ≈ exp 2 T 2 N + o as N , T → ∞ . T 2 Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 → ∞ ? � 1 For δ < 0, φ ( δ, T ) = − π 2 � 2 T 2 + c δ T 3 + o with c δ > 0 explicit. T 3 [Owczarek et al. 2008] prove (for the polymer in a slit) that � N − π 2 � �� Z T N ,δ ≈ exp 2 T 2 N + o as N , T → ∞ . T 2 We show that φ ( δ, T ) can be developed at wish in ( ⋆ ). Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 → ∞ ? � 1 For δ < 0, φ ( δ, T ) = − π 2 � 2 T 2 + c δ T 3 + o with c δ > 0 explicit. T 3 [Owczarek et al. 2008] prove (for the polymer in a slit) that � N − π 2 � �� Z T N ,δ ≈ exp 2 T 2 N + o as N , T → ∞ . T 2 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

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 → ∞ ? � 1 For δ < 0, φ ( δ, T ) = − π 2 � 2 T 2 + c δ T 3 + o with c δ > 0 explicit. T 3 [Owczarek et al. 2008] prove (for the polymer in a slit) that � N − π 2 � �� Z T N ,δ ≈ exp 2 T 2 N + o as N , T → ∞ . T 2 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

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

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Henceforth T N → ∞ . Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Henceforth T N → ∞ . For δ > 0 � positive density of contacts with the interfaces: C δ = ∂φ L N ∼ C δ · N with ∂δ ( δ, ∞ ) > 0. Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Henceforth T N → ∞ . For δ > 0 � positive density of contacts with the interfaces: C δ = ∂φ L N ∼ C δ · N with ∂δ ( δ, ∞ ) > 0. Simple homogeneous pinning model (one interface at zero): max | S i | ≈ log N S N = O ( 1 ) N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Henceforth T N → ∞ . For δ > 0 � positive density of contacts with the interfaces: C δ = ∂φ L N ∼ C δ · N with ∂δ ( δ, ∞ ) > 0. Simple homogeneous pinning model (one interface at zero): T N max | S i | ≈ log N S N = O ( 1 ) N ◮ If T N ≫ log N nothing changes: polymer localized at zero Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Henceforth T N → ∞ . For δ > 0 � positive density of contacts with the interfaces: C δ = ∂φ L N ∼ C δ · N with ∂δ ( δ, ∞ ) > 0. Simple homogeneous pinning model (one interface at zero): T N N ◮ If T N ≫ log N nothing changes: polymer localized at zero ◮ If T N ≪ log N different interfaces worth visiting Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Under P T N τ T N ,δ ( N ≫ 1), how long is the time ˆ 1 to reach level ± T ? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Under P T N τ T N ,δ ( N ≫ 1), how long is the time ˆ 1 to reach level ± T ? 1 ≈ e c δ T with c δ > 0. τ T It turns out that ˆ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Under P T N τ T N ,δ ( N ≫ 1), how long is the time ˆ 1 to reach level ± T ? 1 ≈ e c δ T with c δ > 0. τ T It turns out that ˆ ◮ If e c δ T N ≪ N , there are ≈ N / e c δ T N jumps to a neighboring interface, therefore � √ N e c δ T N = ( e − c δ 2 T N T N ) S N ≍ T N N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Under P T N τ T N ,δ ( N ≫ 1), how long is the time ˆ 1 to reach level ± T ? 1 ≈ e c δ T with c δ > 0. τ T It turns out that ˆ ◮ If e c δ T N ≪ N , there are ≈ N / e c δ T N jumps to a neighboring interface, therefore � √ N e c δ T N = ( e − c δ 2 T N T N ) S N ≍ T N N ◮ If e c δ T N ≈ N , there are O (1) jumps to a neighboring interface, hence S N ≈ T N . Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: heuristics Notation S N ≍ α N means S N /α N is tight and P T N N ,δ ( | S N /α N | ≥ ε ) ≥ ε Under P T N τ T N ,δ ( N ≫ 1), how long is the time ˆ 1 to reach level ± T ? 1 ≈ e c δ T with c δ > 0. τ T It turns out that ˆ ◮ If e c δ T N ≪ N , there are ≈ N / e c δ T N jumps to a neighboring interface, therefore � √ N e c δ T N = ( e − c δ 2 T N T N ) S N ≍ T N N ◮ If e c δ T N ≈ N , there are O (1) jumps to a neighboring interface, hence S N ≈ T N . ◮ If e c δ T N ≫ N , there are no jumps to a neighboring interface, hence S N ≍ O (1). Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N S N = ⇒ N (0 , 1) √ C δ ( e − c δ 2 T N T N ) N √ Note that T N ≪ S N ≪ N. Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N S N = ⇒ N (0 , 1) √ C δ ( e − c δ 2 T N T N ) N √ Note that T N ≪ S N ≪ N. ◮ If T N − 1 c δ log N → ζ ∈ R then S N ≍ T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N S N = ⇒ N (0 , 1) √ C δ ( e − c δ 2 T N T N ) N √ Note that T N ≪ S N ≪ N. ◮ If T N − 1 c δ log N → ζ ∈ R then S N ≍ T N S N = ⇒ S Γ Γ ∼ Poisson( f ( δ, ζ )) T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N S N = ⇒ N (0 , 1) √ C δ ( e − c δ 2 T N T N ) N √ Note that T N ≪ S N ≪ N. ◮ If T N − 1 c δ log N → ζ ∈ R then S N ≍ T N S N = ⇒ S Γ Γ ∼ Poisson( f ( δ, ζ )) T N ◮ If T N − 1 c δ log N → + ∞ then S N ≍ O (1) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results Theorem ( [CP1] ) For every δ > 0 there exists c δ > 0 such that under P T N N ,δ : √ c δ log N → − ∞ then S N ≍ ( e − c δ ◮ If T N − 1 2 T N T N ) N S N = ⇒ N (0 , 1) √ C δ ( e − c δ 2 T N T N ) N √ Note that T N ≪ S N ≪ N. ◮ If T N − 1 c δ log N → ζ ∈ R then S N ≍ T N S N = ⇒ S Γ Γ ∼ Poisson( f ( δ, ζ )) T N ◮ If T N − 1 c δ log N → + ∞ then S N ≍ O (1) P T N � � L →∞ sup lim | S N | > L = 0 N ,δ N ∈ 2 N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The attractive case δ > 0: path results S N √ N 1 c δ log N O ( 1 ) 1 1 c δ log N T N • Sub-diffusive scaling ( T N → ∞ ) • Transition at T N ≈ log N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics Again T N → ∞ . Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics Again T N → ∞ . For δ < 0, zero density of contacts with the interfaces: L N = o ( N ). Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics Again T N → ∞ . For δ < 0, zero density of contacts with the interfaces: L N = o ( N ). Simple homogeneous pinning model (one interface at zero): y S N = O ( N ) 1/2 x N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics Again T N → ∞ . For δ < 0, zero density of contacts with the interfaces: L N = o ( N ). Simple homogeneous pinning model (one interface at zero): y T N S N = O ( N ) 1/2 x N √ √ ◮ If T N ≫ N nothing changes: S N ≍ N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics Again T N → ∞ . For δ < 0, zero density of contacts with the interfaces: L N = o ( N ). Simple homogeneous pinning model (one interface at zero): y T N S N = ? x N √ √ ◮ If T N ≫ N nothing changes: S N ≍ N √ ◮ If T N ≪ N does the polymer visit other interfaces? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics τ T How long is the time ˆ 1 to reach level ± T ? Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics τ T How long is the time ˆ 1 to reach level ± T ? τ T 1 ≈ T 2 ˆ T N 1 ≈ T 2 (diffusivity) τ T Under the simple random walk law ˆ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics τ T How long is the time ˆ 1 to reach level ± T ? τ T 1 ≈ T 3 ˆ T N 1 ≈ T 2 (diffusivity) τ T Under the simple random walk law ˆ 1 ≈ T 3 (repulsion) Under the polymer measure P T τ T N ,δ with δ < 0 ˆ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics τ T How long is the time ˆ 1 to reach level ± T ? τ T 1 ≈ T 3 ˆ T N 1 ≈ T 2 (diffusivity) τ T Under the simple random walk law ˆ 1 ≈ T 3 (repulsion) Under the polymer measure P T τ T N ,δ with δ < 0 ˆ ◮ if T N ≪ N 1 / 3 , there are ≈ N / T 3 N ≫ 1 jumps to a neighboring � N / T 3 � interface, hence S N ≈ T N N = N / T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: heuristics τ T How long is the time ˆ 1 to reach level ± T ? τ T 1 ≈ T 3 ˆ T N 1 ≈ T 2 (diffusivity) τ T Under the simple random walk law ˆ 1 ≈ T 3 (repulsion) Under the polymer measure P T τ T N ,δ with δ < 0 ˆ ◮ if T N ≪ N 1 / 3 , there are ≈ N / T 3 N ≫ 1 jumps to a neighboring � N / T 3 � interface, hence S N ≈ T N N = N / T N √ ◮ if N 1 / 3 ≪ T N ≪ N , there are no jumps to a neighboring interface � confinement: S N ≍ T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 P T N N ,δ : ◮ If T N ≪ N 1 / 3 then S N ≍ � N / T N ≫ T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 P T N N ,δ : ◮ If T N ≪ N 1 / 3 then S N ≍ � N / T N ≫ T N � � S N ≤ P T N � � � � a < Z ≤ b a < ≤ b ≤ c 2 P a < Z ≤ b c 1 P N ,δ � N C δ T N with Z ∼ N (0 , 1) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 P T N N ,δ : ◮ If T N ≪ N 1 / 3 then S N ≍ � N / T N ≫ T N � � S N ≤ P T N � � � � a < Z ≤ b a < ≤ b ≤ c 2 P a < Z ≤ b c 1 P N ,δ � N C δ T N with Z ∼ N (0 , 1) ◮ If T N ∼ ( const . ) N 1 / 3 then S N ≍ T N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 P T N N ,δ : ◮ If T N ≪ N 1 / 3 then S N ≍ � N / T N ≫ T N � � S N ≤ P T N � � � � a < Z ≤ b a < ≤ b ≤ c 2 P a < Z ≤ b c 1 P N ,δ � N C δ T N with Z ∼ N (0 , 1) ◮ If T N ∼ ( const . ) N 1 / 3 then S N ≍ T N √ ◮ If ( const . ) N 1 / 3 ≤ T N ≤ ( const . ) N then S N ≍ T N and P T N � � ∀ ǫ ∃ L : 0 < | S n | < T N , ∀ n ∈ { L , . . . , N } ≥ 1 − ǫ N ,δ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 P T N N ,δ : ◮ If T N ≪ N 1 / 3 then S N ≍ � N / T N ≫ T N � � S N ≤ P T N � � � � a < Z ≤ b a < ≤ b ≤ c 2 P a < Z ≤ b c 1 P N ,δ � N C δ T N with Z ∼ N (0 , 1) ◮ If T N ∼ ( const . ) N 1 / 3 then S N ≍ T N √ ◮ If ( const . ) N 1 / 3 ≤ T N ≤ ( const . ) N then S N ≍ T N and P T N � � ∀ ǫ ∃ L : 0 < | S n | < T N , ∀ n ∈ { L , . . . , N } ≥ 1 − ǫ N ,δ √ √ ◮ If T N ≫ N then S N ≍ N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof The repulsive case δ < 0: path results S N √ N N 1 / 3 √ 1 N 1 / 3 T N N √ √ • Transitions T N ≈ N 1 / 3 , • Sub-diffusive if 1 ≪ T N ≪ N N Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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

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 S n visits an interface τ T n > τ T � � k +1 := inf k : S n − S τ T k ∈ {− T , 0 , T } ( T is fixed) Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 S n visits an interface τ T n > τ T � � k +1 := inf k : S n − S τ 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 (min { n , T 2 } ) 3 / 2 e − π 2 1 2 T 2 n q T ( n ) := P ( τ T 1 = n ) ≈ Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 S n visits an interface τ T n > τ T � � k +1 := inf k : S n − S τ 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 (min { n , T 2 } ) 3 / 2 e − π 2 1 2 T 2 n q T ( n ) := P ( τ T 1 = n ) ≈ Note that under P  O (1) with probab. 1 − 1 1 � � q T ( n ) ≈  T n 3 / 2  τ T ≈ 1   Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

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 S n visits an interface τ T n > τ T � � k +1 := inf k : S n − S τ 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 (min { n , T 2 } ) 3 / 2 e − π 2 1 2 T 2 n q T ( n ) := P ( τ T 1 = n ) ≈ Note that under P  O (1) with probab. 1 − 1 1 � � q T ( n ) ≈  T n 3 / 2  τ T ≈ 1 T 3 e − π 2 � 2 T 2 n � O ( T 2 ) with probab. 1 1 q T ( n ) ≈   T Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof A renewal theory approach Under the polymer measure P T 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

Introduction The Model Free Energy Path Behavior The Proof A renewal theory approach Under the polymer measure P T 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 P δ, T : for both δ > 0 and δ < 0 1 = n ) = e δ P ( τ T K δ, T ( n ) := P δ, T ( τ T 1 = n ) e − φ ( δ, T ) n Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof A renewal theory approach Under the polymer measure P T 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 P δ, T : for both δ > 0 and δ < 0 1 = n ) = e δ P ( τ T K δ, T ( n ) := P δ, T ( τ T 1 = n ) e − φ ( δ, T ) n For δ > 0, we have φ ( δ, T ) → φ ( δ, ∞ ) > 0 as T → ∞ , hence K δ, T ( n ) ≈ e − φ ( δ, ∞ ) n Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010

Introduction The Model Free Energy Path Behavior The Proof A renewal theory approach Under the polymer measure P T 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 P δ, T : for both δ > 0 and δ < 0 1 = n ) = e δ P ( τ T K δ, T ( n ) := P δ, T ( τ T 1 = n ) e − φ ( δ, T ) n For δ > 0, we have φ ( δ, T ) → φ ( δ, ∞ ) > 0 as T → ∞ , hence K δ, T ( n ) ≈ e − φ ( δ, ∞ ) n For δ < 0, we have φ ( δ, T ) ≈ − π 2 2 T 2 + C δ T 3 as T → ∞ , hence  O (1) with probab. e δ 1 � � K δ, T ( n ) ≈  n 3 / 2  τ T ≈ 1   Francesco Caravenna A Polymer in a Multi-Interface Medium December 16, 2010