Random Polymer Models

Transcription

1 Random Polymer Models Disorder and Localization Phenomena Giambattista Giacomin University of Paris 7 Denis Diderot + Probability Lab. (LPMA) Paris 6 & 7 CNRS U.M.A. 7599

2 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.1/65 A Polymer Close to a Membrane exterior more adhesive regions interior Does the polymer stick to the membrane?

3 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.1/65 A Polymer Close to a Membrane exterior more adhesive regions interior Does the polymer stick to the membrane? For localization: contact energy (inhomogeneities?) For delocalization: entropy (fluctuation freedom)

4 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.2/65 A Copolymer Near a Selective Interface + hydrophobic (monomer) hydrophilic Oil Contact Energy Water _ Interface Does the polymer localize at the interface?

5 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.2/65 A Copolymer Near a Selective Interface + hydrophobic (monomer) hydrophilic Oil Contact Energy Water _ Interface Does the polymer localize at the interface? Possible relevant factors (in random order): Distribution of charges (±) Charge asymmetry (quantity or interaction) Polymer intrinsic properties

6 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.3/65 DNA Unbinding (and More) A-T G-C Loop Loop Reduced models (Peyrard-Bishop, Poland-Scheraga)

7 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.3/65 DNA Unbinding (and More) A-T G-C Loop Loop Reduced models (Peyrard-Bishop, Poland-Scheraga) General framework: pinning of chains on defect structures The defect structure is (approximately) linear It may (or may not) be possible to go around the defect

8 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.4/65 A General Mechanism... On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read: "In fact, there is a rather simple but general mathematical mechanism which underlies a broad class of exactly soluble one-dimensional models which display phase transitions. This mechanism does not seem to be as well appreciated as it merits and it operates in a number of applications we wish to discuss."

9 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.4/65 A General Mechanism... On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read: "In fact, there is a rather simple but general mathematical mechanism which underlies a broad class of exactly soluble one-dimensional models which display phase transitions. This mechanism does not seem to be as well appreciated as it merits and it operates in a number of applications we wish to discuss." Fisher s broad class of models is precisely the class we are focusing on if we limit ourselves to homogeneous models (and we start from there). We aim at treating inhomogeneous (disordered) models.

10 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.4/65 A General Mechanism... On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read: "In fact, there is a rather simple but general mathematical mechanism which underlies a broad class of exactly soluble one-dimensional models which display phase transitions. This mechanism does not seem to be as well appreciated as it merits and it operates in a number of applications we wish to discuss." G.G., Random Polymer Models, IC press (to appear) From my web-page research (.../pub/rpmay30.pdf.gz)

11 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.5/65 The Basic (Pinning) Model Free process: Symmetric Random Walk {S n } n S 0 = 0, S n = n j=1 X j, {X j } j IID with P (X 1 = x) = 1/3, x { 1, 0, +1}.

12 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.5/65 The Basic (Pinning) Model Free process: Symmetric Random Walk {S n } n S 0 = 0, S n = n j=1 X j, {X j } j IID with P (X 1 = x) = 1/3, x { 1, 0, +1}. Coupling parameter: β R dp c N,β dp (S) = 1 Z c N,β exp ( β N n=1 1 Sn =0 ) 1 SN =0.

13 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.6/65 The Basic (Pinning) Model S n 0 N n In this case: N = 16 and the energetic contribution is 6β. dp c N,β dp (S) = 1 Z c N,β exp ( β N n=1 1 Sn =0 ) 1 SN =0.

14 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.7/65 The Basic Pinning Model Notation and facts: τ 0 = 0 and τ n = inf{m > τ n 1 : S m = 0} if τ n 1 < (τ n = otherwise).

15 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.7/65 The Basic Pinning Model Notation and facts: τ 0 = 0 and τ n = inf{m > τ n 1 : S m = 0} if τ n 1 < (τ n = otherwise). By the (strong) Markov property η := {η i } i N, η i = τ i τ i 1, is IID.

16 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.7/65 The Basic Pinning Model Notation and facts: τ 0 = 0 and τ n = inf{m > τ n 1 : S m = 0} if τ n 1 < (τ n = otherwise). By the (strong) Markov property η := {η i } i N, η i = τ i τ i 1, is IID. K(n) := P(τ 1 = n). Well known that K(n) n C K n 3/2 (C K > 0). Moreover n K(n) = 1 (i.e. S is recurrent).

17 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.7/65 The Basic Pinning Model Notation and facts: τ 0 = 0 and τ n = inf{m > τ n 1 : S m = 0} if τ n 1 < (τ n = otherwise). By the (strong) Markov property η := {η i } i N, η i = τ i τ i 1, is IID. K(n) := P(τ 1 = n). Well known that K(n) n C K n 3/2 (C K > 0). Moreover n K(n) = 1 (i.e. S is recurrent). For us τ := {τ n } n is also a random subset of N. So expressions like N τ and τ A are meaningful.

18 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.8/65 The Basic Pinning Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β 0, and F(β) := 0 otherwise.

19 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.8/65 The Basic Pinning Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β 0, and F(β) := 0 otherwise. Note that: F(β) > 0 for β > 0. F : R [0, ) is non decreasing and C 0 (in fact: analytic on R \ {0}: β c = 0).

20 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.8/65 The Basic Pinning Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β 0, and F(β) := 0 otherwise. Proposition 1. For every β F(β) = lim N 1 N log Zc N,β.

21 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.9/65 Proof of Prop. 1 The fundamental formula is Z c N,β = N n=1 l N n : P n j=1 l j=n n j=1 exp(β)k(l j ).

22 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.9/65 Proof of Prop. 1 The fundamental formula is Z c N,β = N n=1 l N n : P n j=1 l j=n n j=1 exp(β)k(l j ). S 0 N l 1 l 2 l 3 l 4 l 5 l 6

23 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.9/65 Proof of Prop. 1 The fundamental formula is Z c N,β = N n=1 l N n : P n j=1 l j=n n j=1 exp(β)k(l j ). Set K β (n) := exp(β)k(n) exp( F(β)n). Then Z c N,β = exp (F(β)N) N n=1 l N n : P n j=1 l j=n n j=1 K β (l j ).

24 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.9/65 Proof of Prop. 1 The fundamental formula is Z c N,β = N n=1 l N n : P n j=1 l j=n n j=1 exp(β)k(l j ). Set K β (n) := exp(β)k(n) exp( F(β)n). Then Z c N,β = exp (F(β)N) N n=1 l N n : P n j=1 l j=n n j=1 K β (l j ).

25 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.10/65 Proof of Prop. 1 With P β the law of the renewal τ on N { } with inter-arrival law K β ( ): Z c N,β = exp(f(β)n) P β (N τ),

26 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.10/65 Proof of Prop. 1 With P β the law of the renewal τ on N { } with inter-arrival law K β ( ): Z c N,β = exp(f(β)n) P β (N τ), so Z c N,β exp (F(β)N), and it suffices to show that lim inf N 1 N log P β (N τ) 0.

27 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.11/65 Proof of Prop. 1 If β > β c : m ekβ := n n K β (n) < and (by the Renewal Theorem) P β (N τ) N 1/m ekβ > 0.

28 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.11/65 Proof of Prop. 1 If β > β c : m ekβ := n n K β (n) < and (by the Renewal Theorem) P β (N τ) N 1/m ekβ > 0. If β < β c : use P β (N τ) K β (N) = exp(β)k(n), which is bounded below by const./n 3/2.

29 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.11/65 Proof of Prop. 1 If β > β c : m ekβ := n n K β (n) < and (by the Renewal Theorem) P β (N τ) N 1/m ekβ > 0. If β < β c : use P β (N τ) K β (N) = exp(β)k(n), which is bounded below by const./n 3/2. If β = β c : Kβ ( ) = K( ) and P β (N τ) = P(N τ) K(N) const./n 3/2.

30 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.11/65 Proof of Prop. 1 If β > β c : m ekβ := n n K β (n) < and (by the Renewal Theorem) P β (N τ) N 1/m ekβ > 0. If β < β c : use P β (N τ) K β (N) = exp(β)k(n), which is bounded below by const./n 3/2. If β = β c : Kβ ( ) = K( ) and P β (N τ) = P(N τ) K(N) const./n 3/2. The proof is complete.

31 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.12/65 A Further Look at the Proof We have in fact proven the sharp estimate Z c N,β = (1 + o(1)) m ekβ exp (N F(β)), for every β.

32 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.12/65 A Further Look at the Proof We have in fact proven the sharp estimate Z c N,β = (1 + o(1)) m ekβ exp (N F(β)), for every β. This can be improved for β < β c Z c N,β = (c β + o(1))n 3/2, and at criticality: Z c N,β c = (c 0 + o(1))n 1/2. We ll come back to this.

33 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.13/65 What about the trajectories? By convexity the contact fraction is: β F(β) = lim N 1 N β log Z c N,β = lim N 1 N Ec N,β [ N n=1 1 Sn =0 ], if β F(β) exists.

34 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.13/65 What about the trajectories? By convexity the contact fraction is: β F(β) = lim N 1 N β log Z c N,β = lim N 1 N Ec N,β [ N n=1 1 Sn =0 ], if β F(β) exists. Therefore: The contact fraction is zero for β < β c. The contact fraction is positive for β > β c. So the localization can be characterized by F(β) > 0.

35 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.14/65 A basic (penetrable) wall model Coupling parameter: h [0, ] dp c,+ N,λ dp (S) = 1 Z c,+ N,h exp ( h N n=1 1 Sn <0 ) 1 SN =0.

36 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.14/65 A basic (penetrable) wall model Coupling parameter: h [0, ] dp c,+ N,λ dp (S) = 1 Z c,+ N,h exp ( h N n=1 1 Sn <0 ) 1 SN =0. One sees immediately that F(h) = 0 and that h F(h) = lim N 1 N Ec,+ N,h [ N n=1 1 Sn <0 ], so the walk spends (essentially) all is time in the upper half plane. Keyword: Entropic Repulsion.

37 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.15/65 A basic (penetrable) wall model For a closer look let us write Z c,+ N,h = N n=1 l N n : P n j=1 l j=n n j=1 1 2 (exp( h(l j 1)) + 1)K(l j ).

38 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.15/65 A basic (penetrable) wall model For a closer look let us write Z c,+ N,h = N n=1 l N n : P n j=1 l j=n n 1 2 (exp( h(l j 1)) + 1)K(l j ). }{{} j=1 If we set K h (l j ) and h > 0 we have n K h(n) < 1.

39 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.15/65 A basic (penetrable) wall model For a closer look let us write Z c,+ N,h = N n=1 l N n : P n j=1 l j=n n 1 2 (exp( h(l j 1)) + 1)K(l j ). }{{} j=1 If we set K h (l j ) and h > 0 we have n K h(n) < 1. This strongly suggests that, as N, P c,+ N,h converges to the terminating renewal τ with inter-arrival distribution K h ( ) and results like (L large) lim N Pc,+ N,h (τ (L,N L) = ) = 1 o L(1).

40 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.16/65 The General Homogeneous Model If τ is a general renewal with inter-arrivals taking values in N we introduce N N (τ) = τ {1, 2,...,N} and dp c N,β dp (τ) = 1 Z c N,β exp (βn N (τ))1 N τ.

41 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.16/65 The General Homogeneous Model If τ is a general renewal with inter-arrivals taking values in N we introduce N N (τ) = τ {1, 2,...,N} and dp c N,β dp (τ) = 1 Z c N,β exp (βn N (τ))1 N τ. We focus on inter-arrivals with law K(n) = L(n) n1+α, n = 1,2,... with L( ) slowly varying and α 0. We set Σ K := n N K(n), K( ) := 1 Σ K and m K := n N { } nk(n).

42 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.17/65 The General Homogeneous Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β β c = log Σ K, and F(β) := 0 otherwise.

43 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.17/65 The General Homogeneous Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β β c = log Σ K, and F(β) := 0 otherwise. Therefore: F(β) > 0 for β > β c. F : R [0, ) is non decreasing and C 0 (in fact: analytic on R \ {β c }).

44 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.17/65 The General Homogeneous Model Call F(β) the only solution of K(n) exp( F(β)n) = exp( β), n when such a solution exists, that is for β β c = log Σ K, and F(β) := 0 otherwise. Proposition 2. For every β F(β) = lim N 1 N log Zc N,β.

45 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.18/65 The General Homogeneous model Critical behavior: Theorem 3. For every α 0 and every L( ) there exists a slowly varying function ˆL( ) such that for δ > 0 F(β c + δ) = δ 1/min(1,α) ˆL(1/δ). In particular if n N nk(n) < then lim δց0 ˆL(1/δ) = ΣK / n N nk(n).

46 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.18/65 The General Homogeneous model Critical behavior: Theorem 3. For every α 0 and every L( ) there exists a slowly varying function ˆL( ) such that for δ > 0 F(β c + δ) = δ 1/min(1,α) ˆL(1/δ). In particular if n N nk(n) < then lim δց0 ˆL(1/δ) = ΣK / n N nk(n). Therefore the transition is of k th order if α (1/k,1/(k 1)).

47 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.19/65 The General Homogeneous model F(β) N(β) α > 1 1 β c 0 1 β

48 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.20/65 The General Homogeneous Model Sharp estimates on the partition function [Deuschel, G., Zambotti PTRF2005], [Caravenna, G., Zambotti, EJP2006]: Theorem 4. As N we have: 1. (The localized regime.) If β > β c then Z c N,β 1 m ekβ exp(f(β)n). 2. (The strictly delocalized regime.) If β < β c then Z c N,β exp(β) (1 ) 2 K(N), where := exp(β)σ K (< 1).

49 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.21/65 The General Homogeneous model Theorem 4 (Cont.) 3. (The critical regime. ) If we set K(n) := j>n K(j) and m N := N n=0 K(n) then N m N is increasing and, if α 1, slowly varying and { ZN,β c 1/m N if α 1, c Σ K α sin(πα)n α 1 /(πl(n)) if α (0, 1). If P n nk(n) < then m N may be substituted by m. If K( ) = 0 then m is the mean of the inter-arrival. Subtle result in the critical regime [R. A. Doney (1997)]. Open for α = 0.

50 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.22/65 The General Homogeneous model Why are we so much interested in sharp estimates on Z c N,β?

51 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.22/65 The General Homogeneous model Why are we so much interested in sharp estimates on Z c N,β? Sharp estimates ( =) = Sharp path behavior

52 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.22/65 The General Homogeneous model Why are we so much interested in sharp estimates on Z c N,β? Sharp estimates ( =) = Sharp path behavior { } Proposition 5. For every β the sequence PN,β c converges N weakly to P β, the law of the renewal τ β with inter-arrival Kβ ( ). τ β is positive recurrent if β > β c. τ β is terminating if β < β c. τ β is recurrent if β = β c.

53 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.23/65 The General Homogeneous Model Proof of Prop. 5: l 1,l 2 N and q := P c N,β (τ 1 = l 1,τ 2 = l 2 ) q = eβ K(l 1 )e β K(l 2 )Z c N l 1 l 2,β Z c N,β = K β (l 1 ) K β (l 2 )R N, with R N := exp((l 1 + l 2 )F(β)) Zc N l 1 l 2,β Z c N,β = P β (N l 1 l 2 τ) P β (N τ), and R N N 1 by the sharp estimates. Scaling limits: τ (N) := τ N [0,1]. Particularly interesting: critical case.

54 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.24/65 Back to the "General Mechanism" M. E. Fisher s approach considers the Laplace transform of ZN,β c : Θ(µ) := exp( µn)zn,β c, N which can be computed explicitly. By super-additive arguments it is easy to show that lim N (1/N) log ZN,β c exists and this immediately implies that this limit coincides with inf {µ : Θ(µ) < }.

55 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.24/65 Back to the "General Mechanism" M. E. Fisher s approach considers the Laplace transform of ZN,β c : Θ(µ) := exp( µn)zn,β c, N which can be computed explicitly. By super-additive arguments it is easy to show that lim N (1/N) log ZN,β c exists and this immediately implies that this limit coincides with inf {µ : Θ(µ) < }. Extracting sharp estimates requires Tauberian arguments. [Erdös, Pollard, Feller, Port, Garsia, Lamperti,...(1949 onwards)]

56 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.25/65 A Copolymer Near a Selective Interface + hydrophobic (monomer) hydrophilic Oil Contact Energy Water _ Interface Does the polymer localize at the interface?

57 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.25/65 A Copolymer Near a Selective Interface + hydrophobic (monomer) hydrophilic Oil Contact Energy Water _ Interface Does the polymer localize at the interface? Possible relevant factors (in random order): Distribution of charges (±) Charge asymmetry (quantity or interaction) Polymer intrinsic properties

58 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.26/65 Weakly Inhomogeneous Models Periodic charge sequence ω := {ω n } n : for some T N and every n. ω n R and ω n+t = ω n,

59 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.26/65 Weakly Inhomogeneous Models Periodic charge sequence ω := {ω n } n : for some T N and every n. ω n R and ω n+t = ω n, A copolymer model: S is the simple random walk (N 2N), λ,h 0, ω n = ( 1) n (so T = 2) and sign(0) = +1 dp c N,ω,λ,h dp (S) = 1 Z c N,ω,β exp ( λ ) N (ω n + h)sign (S n ) n=1 [Garel, Monthus, Orland (2000)], [Sommer, Daoud (1996)],... 1 SN =0.

60 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.27/65 Weakly Inhomogeneous Copolymer S n n λ P N n=1 (ω n + h) sign(s n ), sign(0) = +1 +

61 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.28/65 Weakly Inhomogeneous Copolymer Observe that: N λ (ω n + h)sign (S n ) = n=1 N N 2λ (ω n + h)1 sign(sn = 1) + λ (ω n + h). n=1 n=1 So: the copolymer is equivalent to a pinning model with exp( β) := n 1 + exp(+2λ 2λhn) K(n), 2 and K(n) replaced by the probability e K(n) := exp( e β) 1+exp(+2λ 2λhn) 2 K(n).

62 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.29/65 Weakly Inhomogeneous Copolymer From this computation we extract the phase diagram: h D [GMO]-model L 0 [SD]-model λ

63 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.30/65 Weakly Inhomogeneous Copolymer More general case: T > 2. Periodic pinning and periodic copolymer interaction.

64 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.30/65 Weakly Inhomogeneous Copolymer More general case: T > 2. Periodic pinning and periodic copolymer interaction. Possible strategy of solution: decimation (of step T ). Problem: the model does not renormalize to a pinning.

65 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.30/65 Weakly Inhomogeneous Copolymer More general case: T > 2. Periodic pinning and periodic copolymer interaction. Possible strategy of solution: decimation (of step T ). Problem: the model does not renormalize to a pinning. [Bolthausen, G. AAP2005]: an expression for the free energy (LD arguments). [Caravenna, G., Zambotti2005,2006]: Renewal theory approach (sharp estimates and path behavior). The underlying contact site process is a Markov renewal richer phenomenology.

66 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions.

67 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions. Free energy, localization and delocalization.

68 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions. Free energy, localization and delocalization. Path properties (results and open problems).

69 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions. Free energy, localization and delocalization. Path properties (results and open problems). Free energy estimates (rare stretch strategy).

70 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions. Free energy, localization and delocalization. Path properties (results and open problems). Free energy estimates (rare stretch strategy). A smoothing mechanism.

71 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.31/65 Strongly Inhomogeneous Models Program: Quenched disordered models: definitions. Free energy, localization and delocalization. Path properties (results and open problems). Free energy estimates (rare stretch strategy). A smoothing mechanism. A number of questions can be treated for copolymers and pinning at the same time. But on some other issues pinning and copolymer models have sharply different behaviors.

72 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.32/65 Disordered Models: Definition Free process: (τ, s) couple of independent processes s.t. τ renewal process on N { } with inter-arrival law K( ) (K(n) = L(n)/n 1+α ); s = {s j } j N IID with P(s 1 = ±1) = 1/2; S: walk s.t. {n N {0} : S n = 0} = τ \ { } and sign(s n ) = s τj, j = inf{i : τ i n}.

73 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.32/65 Disordered Models: Definition Free process: (τ, s) couple of independent processes s.t. τ renewal process on N { } with inter-arrival law K( ) (K(n) = L(n)/n 1+α ); s = {s j } j N IID with P(s 1 = ±1) = 1/2; S: walk s.t. {n N {0} : S n = 0} = τ \ { } and sign(s n ) = s τj, j = inf{i : τ i n}. Parameters: v Charges ω R N (centered quenched disorder) dp c N,ω,v dp (S) = 1 Z c N,ω,v ) exp (H v N,ω (S) 1 {SN =0}.

74 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.32/65 Disordered Models: Definition Free process: (τ, s) couple of independent processes s.t. τ renewal process on N { } with inter-arrival law K( ) (K(n) = L(n)/n 1+α ); s = {s j } j N IID with P(s 1 = ±1) = 1/2; S: walk s.t. {n N {0} : S n = 0} = τ \ { } and sign(s n ) = s τj, j = inf{i : τ i n}. Parameters: v Charges ω R N (centered quenched disorder) dp f N,ω,v dp (S) = 1 Z c N,ω,v ) exp (H v N,ω (S).1 {SN =0}

75 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.33/65 Disordered Models: Definition Copolymer: v = (λ,h), λ,h 0 H λ,h N,ω (S) = λ N n=1 (ω n + h)sign (S n )

76 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.33/65 Disordered Models: Definition Copolymer: v = (λ,h), λ,h 0 H λ,h N,ω (S) = λ N n=1 (ω n + h)sign (S n ) = λ N N (τ) j=1 s j τ j n=τ j 1 +1 (ω n + h).

77 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.33/65 Disordered Models: Definition Copolymer: v = (λ,h), λ,h 0 H λ,h N,ω (S) = λ N n=1 (ω n + h)sign (S n ) = λ N N (τ) j=1 s j τ j n=τ j 1 +1 (ω n + h). Pinning at a defect line: v = (β,h), β 0, h R H β,h N,ω (S) = N n=1 (βω n h)1 {Sn =0}

78 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.33/65 Disordered Models: Definition Copolymer: v = (λ,h), λ,h 0 H λ,h N,ω (S) = λ N n=1 (ω n + h)sign (S n ) = λ N N (τ) j=1 s j τ j n=τ j 1 +1 (ω n + h). Pinning at a defect line: v = (β,h), β 0, h R H β,h N,ω (S) = N n=1 (βω n h)1 {Sn =0} = n τ: 1 n N (βω n h).

79 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.34/65 Choice of the disorder Random (quenched): {ω n } n IID (law P) such that M(t) := E[exp(tω 1 )] < for t small E [ω 1 ] = 0 (let us say ω 1 ω 1 ) Without loss of generality: E [ ω 2 ] 1 = 1.

80 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.34/65 Choice of the disorder Random (quenched): {ω n } n IID (law P) such that M(t) := E[exp(tω 1 )] < for t small E [ω 1 ] = 0 (let us say ω 1 ω 1 ) Without loss of generality: E [ ω 1 2 ] = 1. Copolymer: λ,h 0, n := (1 sign(s n ))/2 {0,1} H λ,h N,ω (S) = 2λ N n=1 (ω n + h) n + c N (ω)

81 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.35/65 The Free Energy { Theorem 5. The limit of the sequence of r.v. s 1N log ZN,ω,v a N exists P( dω) a.s. and in L 1 (P), it is P( dω) a.s. equal to a constant (self-averaging!) that we denote by F(v). }

82 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.35/65 The Free Energy { Theorem 5. The limit of the sequence of r.v. s 1N log ZN,ω,v a N exists P( dω) a.s. and in L 1 (P), it is P( dω) a.s. equal to a constant (self-averaging!) that we denote by F(v). Several "different" proofs: for example observe that } log Z c N,ω log Z c M,ω + log Z c N M,θ M ω for M = 0,..., N. So n o E log ZN,ω c N is superadditive, so lim N 1 N Elog Zc N,ω = sup N A concentration bound suffices to conclude. 1 N Elog Zc N,ω.

83 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.36/65 Free energy approach The free energy: lim N 1 N log Zf N,ω,v = lim N P( dω) a.s. and in L 1 (P). 1 [ ( )] N loge exp H λ,h N,ω (S) = F(λ,h) Identification of the free energy of the delocalized state: E h i exp H λ,h N,ω (S) " E exp 2λ! NX (ω n + h) n n=1 ; S n > 0 for n = 1,2,..., N #

84 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.36/65 Free energy approach The free energy: lim N 1 N log Zf N,ω,v = lim N P( dω) a.s. and in L 1 (P). 1 [ ( )] N loge exp H λ,h N,ω (S) = F(λ,h) Identification of the free energy of the delocalized state: E h i exp H λ,h N,ω (S) E " P( ) 1/N α # z } { 1 ; S n > 0 for n = 1,2,..., N = exp(o(n))

85 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.36/65 Free energy approach The free energy: lim N 1 N log Zf N,ω,v = lim N P( dω) a.s. and in L 1 (P). 1 [ ( )] N loge exp H λ,h N,ω (S) = F(λ,h) 0 Identification of the free energy of the delocalized state: E h i exp H λ,h N,ω (S) E " P( ) 1/N α # z } { 1 ; S n > 0 for n = 1,2,..., N = exp(o(n))

86 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.37/65 Localized and Delocalized regions D = {v : F(v) = 0} L = {v : F(v) > 0}

87 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.37/65 Localized and Delocalized regions D = {v : F(v) = 0} L = {v : F(v) > 0} The analysis (not obvious!): Understanding the phase diagram What does v L (or D) mean for the polymer trajectories?

88 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.37/65 Localized and Delocalized regions D = {v : F(v) = 0} L = {v : F(v) > 0} The analysis (not obvious!): Understanding the phase diagram What does v L (or D) mean for the polymer trajectories? Chronological math approach: Path analysis free energy approach [S]:=[Sinai TPA93] [BdH97]:=[Bolthausen, den Hollander AP97]

89 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.38/65 Pinning: Phase Diagram h Energy= P n τ [1,N] (βω n h) D h c (0) + log M( ) h c ( ) 0 β h c (0) L h c (0) + h LB ( ) There exists h c ( ) (convex, increasing) separating L and D. The upper bound is the annealed bound. h LB ( ) 0 is easy. h LB ( ) > 0: [Alexander, Sidoravicius 05], [G]

90 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.39/65 Copolymer: Phase Diagram [S], [BdH], [BG]:=[Bodineau and G. JSP04], [GT05]:=[G., Toninelli PTRF05] h c (λ) D L 0 Energy= λ N n=1 (ω n + h)1 sign(sn )= 1 λ

91 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.39/65 Copolymer: Phase Diagram [S], [BdH], [BG]:=[Bodineau and G. JSP04], [GT05]:=[G., Toninelli PTRF05] ( P n K(n) = 1) h c (λ) D L 0 λ As λ ց 0: h c (λ) = m c λ + o(λ), m c [1/(1 + α),1]. Recently: m c 1/ 1 + α if α 1.

92 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.40/65 Phase diagram estimates h (m) (λ) := 1 2mλ log M (2mλ), h(1/(1+α)) (λ) h c (λ) h (1) (λ) h h c(0) [1/(1 + α),1] h (1) (λ) h c (λ) h (1/(1+α)) (λ) 0 λ

93 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.41/65 Phase diagram estimates The physical literature is split: α = 1/2 and L(n) n c > 0 For h (1) ( ) = h c ( ): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case) h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

94 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.41/65 Phase diagram estimates The physical literature is split: α = 1/2 and L(n) n c > 0 For h (1) ( ) = h c ( ): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case) For h (2/3) ( ) = h c ( ): [Stepanov et al. 98] (Replica, Gauss) [Monthus 00] (Fisher s renormalization scheme) h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

95 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.42/65 What Does the Computer Say? [Caravenna, G., Gubinelli JSP06] Statistical test based on concentration and super-additivity: h c ( ) > h (2/3) ( ) h c(0) 0.83 ± 0.01 [Garel, Monthus 05] h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

96 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.43/65 Testing for Localization Basic ingredients: Super-additivity of { } Elog ZN,ω,v c implies: N v L there exists N s.t. Elog Z c N,ω,v > 0.

97 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.43/65 Testing for Localization Basic ingredients: Super-additivity of { } Elog ZN,ω,v c implies: N v L there exists N s.t. Elog ZN,ω,v c > 0. For a wide class of r.v. s ω 1 we have the concentration bound: there exist c 1,c 2 > 0 such that for every convex G : R k R satisfying G(r) G(r ) C r r we have that for every t P( G(ω) EG(ω) t) c 1 exp ( c 2 t 2 /C 2).

98 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.44/65 Testing for Localization In our case k = nn and G(ω) := 1 n P n j=1 log Zc N,θ j 1 ω. Now C2 = 16λ 2 N/n and 0 1 n nx i log Z hlog c N,θ j 1 ω E ZN,ω c j=1 1 ta c 1 exp ` c 2 nt 2 /(16λ 2 N).

99 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.44/65 Testing for Localization In our case k = nn and G(ω) := 1 n P n j=1 log Zc N,θ j 1 ω. Now C2 = 16λ 2 N/n and 0 1 n nx i log Z hlog c N,θ j 1 ω E ZN,ω c j=1 1 ta c 1 exp ` c 2 nt 2 /(16λ 2 N). Therefore: fix N and make the hypothesis that E[log ZN,ω c ] 0. Compute n independent samples of log ZN,ω c (comp. time O(nN 3/2 )) and compute the mean û of the results. If û > 0 we refuse the hypothesis (therefore: localization!) with a level of error smaller than c 1 exp ( c 2 nû 2 /(16λ 2 N) ).

100 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.45/65 A quick look at the paths Real question: what do the paths do? The localized phase Large Deviations regime [S], [Albeverio, Zhou JSP96] [Biskup, den Hollander AAP99] Gibbs measure viewpoint [G., Toninelli Alea06] two distinct correlation lengths

101 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.45/65 A quick look at the paths Real question: what do the paths do? The localized phase Large Deviations regime [S], [Albeverio, Zhou JSP96] [Biskup, den Hollander AAP99] Gibbs measure viewpoint [G., Toninelli Alea06] two distinct correlation lengths The delocalized phase Moderate Deviations If v D = zero density (o(n)) of visits to the interface

102 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.45/65 A quick look at the paths Real question: what do the paths do? The localized phase Large Deviations regime [S], [Albeverio, Zhou JSP96] [Biskup, den Hollander AAP99] Gibbs measure viewpoint [G., Toninelli Alea06] two distinct correlation lengths The delocalized phase Moderate Deviations If v D = zero density (o(n)) of visits to the interface [G., Toninelli PTRF05]: O(log N) and O(1) results If v D: O(N 2/3 ) visits to the interface [Toninelli JSP06]

103 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.45/65 A quick look at the paths Real question: what do the paths do? The localized phase Large Deviations regime [S], [Albeverio, Zhou JSP96] [Biskup, den Hollander AAP99] Gibbs measure viewpoint [G., Toninelli Alea06] two distinct correlation lengths The delocalized phase Moderate Deviations If v D = zero density (o(n)) of visits to the interface [G., Toninelli PTRF05]: O(log N) and O(1) results If v D: O(N 2/3 ) visits to the interface [Toninelli JSP06] Expected in D (?): scaling toward Brownian meander

104 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.46/65 On the Delocalized phase [GT05] Theorem 6. Besides the standard assumptions on ω, let us assume that there exist positive constants c 1 and c 2 s.t. P(G(ω 1,...,ω k ) EG(ω 1,...,ω k ) > t) c 1 exp ( c 2 t 2), for every t 0 and every (convex) G : R k R such that G(x) G(y) x y. Then for every (β,h) D there exist two positive constants c and q such that EP a N,ω (N N (τ) n) exp( cn), both for a = c and a = f and for every N and n q log N.

105 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.47/65 Proof of Theorem 6 Set F a N,ω(β,h;n) := 1 N log Za N,ω (N N (τ) = n), and F a N,ω (β,h) := (1/N)log Za N,ω.

106 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.47/65 Proof of Theorem 6 Set F a N,ω(β,h;n) := 1 N log Za N,ω (N N (τ) = n), and F a N,ω (β,h) := (1/N)log Za N,ω. Important because: P c N,ω (N N (τ) m) = 1 Z c N,ω n m Z c N,ω (N N (τ) = n).

107 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.47/65 Proof of Theorem 6 Set F a N,ω(β,h;n) := 1 N log Za N,ω (N N (τ) = n), and F a N,ω (β,h) := (1/N)log Za N,ω. We have: Lemma 7. Same assumptions as in Th. 6. We have P ( F a N,ω(β,h;n) E [ F a N,ω(β,h;n) ] u ) c 1 exp ( c 2u 2 N 2 ) β 2, n for every N N, n {0,1,...,N} and every u 0.

108 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.47/65 Proof of Theorem 6 Set F a N,ω(β,h;n) := 1 N log Za N,ω (N N (τ) = n), and F a N,ω (β,h) := (1/N)log Za N,ω. We have: Lemma 7. Same assumptions as in Th. 6. We have P ( F a N,ω(β,h;n) E [ F a N,ω(β,h;n) ] u ) c 1 exp ( c 2u 2 N 2 ) β 2, n for every N N, n {0,1,...,N} and every u 0. If n = N we recover the standard fluctuation/concentration bound, but for small n the bound improves substantially.

109 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.48/65 Proof of Theorem 6 Choose (β,h) D and ε > 0 s.t. (β,h ε) D. Super-additivity guarantees: (β,h) D E [ F c N,ω(β,h) ] 0 for every N N.

110 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.48/65 Proof of Theorem 6 Choose (β,h) D and ε > 0 s.t. (β,h ε) D. Super-additivity guarantees: (β,h) D E [ F c N,ω(β,h) ] 0 for every N N. For every ω we have the equivalence F c N,ω(β,h;n) 1 εn/n 2 Fc N,ω(β,h ε;n) 1 2 εn/n, and F c N,ω (β,h ε;n) Fc N,ω (β,h ε).

111 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.48/65 Proof of Theorem 6 Choose (β,h) D and ε > 0 s.t. (β,h ε) D. Super-additivity guarantees: (β,h) D E [ F c N,ω(β,h) ] 0 for every N N. For every ω we have the equivalence F c N,ω(β,h;n) 1 εn/n 2 Fc N,ω(β,h ε;n) 1 2 εn/n, and F c N,ω (β,h ε;n) Fc N,ω (β,h ε). Note that EF c N,ω (β,h ε) 0.

112 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.49/65 Proof of Theorem 6 Therefore (Lemma 7) «P F cn,ω (β, h; n) 12 εn/n i P F cn,ω hf (β, h ε; n) E cn,ω (β, h ε; n) 12 «εn/n «! ε 2 c 1 exp c 2 n. 2 β

113 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.49/65 Proof of Theorem 6 Therefore (Lemma 7) «P F cn,ω (β, h; n) 12 εn/n i P F cn,ω hf (β, h ε; n) E cn,ω (β, h ε; n) 12 «εn/n «! ε 2 c 1 exp c 2 n. 2 β With E m := {ω : n m s.t. F c N,ω (β, h; n) εn/2n}, we have P(E m ) X «P F cn,ω (β, h; n) 12 εn/n n m C 1 1 exp( C 1 m), for every m and some C 1 > 0.

114 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.50/65 Proof of Theorem 6 Choose now ω E m. We have (α > α + ): P c N,ω (N N(τ) m) = 1 Z c N,ω n m Z c N,ω (N N(τ) = n) C 2 N 1+α + exp ( βω N + h) n m exp( εn/2) C 3 N 1+α + exp ( βω N + h) exp ( εm/2), and by taking the expectation we conclude.

115 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.51/65 Localization and Smoothing Program: Copolymer model: lower bound on critical curve. All models: smoothing inequality.

116 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.51/65 Localization and Smoothing Program: Copolymer model: lower bound on critical curve. All models: smoothing inequality. Common idea: role of rare stretches of charges

117 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.52/65 Phase diagram estimates h (m) (λ) := 1 2mλ log M (2mλ), h(1/(1+α)) (λ) h c (λ) h (1) (λ) h h c(0) [1/(1 + α),1] h (1) (λ) h c (λ) h (1/(1+α)) (λ) 0 λ

118 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.53/65 Phase diagram estimates The physical literature is split: α = 1/2 and L(n) n c > 0 For h (1) ( ) = h c ( ): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case) h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

119 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.53/65 Phase diagram estimates The physical literature is split: α = 1/2 and L(n) n c > 0 For h (1) ( ) = h c ( ): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case) For h (2/3) ( ) = h c ( ): [Stepanov et al. 98] (Replica, Gauss) [Monthus 00] (Fisher s renormalization scheme) h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

120 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.54/65 What Does the Computer Say? [Caravenna, G., Gubinelli JSP06] Statistical test based on concentration and super-additivity: h c ( ) > h (2/3) ( ) h c(0) 0.83 ± 0.01 [Garel, Monthus 05] h h (1) (λ) h c (λ) h (2/3) (λ) 0 λ

121 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.55/65 Rare Stretch Strategy We want to prove: h (m) (λ) := 1 2mλ log M (2mλ), h(1/(1+α)) (λ) h c (λ) (BG04) Picture to keep in mind: N and then l S n Atypical ω n s 0 l N n

122 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.56/65 Proof of inequality (BG04) S n Q 1 Q 2 Q j ql 0 G 1 l/p(l) l G 2 l/p(l) l N n with Q j := jl n=(j 1)l+1 (ω j + h). Note that {Q j } j is IID and p(l) := P(Q 1 ql) l exp( lσ( q h)), with Σ( ) the Cramér LD functional. {G j } j IID too. We restrict Z N,ω to the trajectories on the figure (lower bound).

123 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.57/65 Proof of inequality (BG04) In one excursion the contribution to log Z N,ω is: log K ( G j ) l +log (K(l) exp(2λql)) (1+α)Σ( q h)l+2λql. p(l)

124 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.57/65 Proof of inequality (BG04) In one excursion the contribution to log Z N,ω is: log K ( G j ) l +log (K(l) exp(2λql)) (1+α)Σ( q h)l+2λql. p(l) We can still optimize over q: we gain if 0 < gain := sup q (2λq (1 + α)σ( q h)) = 2λh + (1 + α)sup q (( 2λ/(1 + α))q Σ(q)) = 2λh + (1 + α)log M ( 2λ/(1 + α)), obtaining, for l sufficiently large, log Z N,ω Np(l)gain/2.

125 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.57/65 Proof of inequality (BG04) In one excursion the contribution to log Z N,ω is: log K ( G j ) l +log (K(l) exp(2λql)) (1+α)Σ( q h)l+2λql. p(l) We can still optimize over q: we gain if 0 < gain := sup q (2λq (1 + α)σ( q h)) = 2λh + (1 + α)sup q (( 2λ/(1 + α))q Σ(q)) = 2λh + (1 + α)log M ( 2λ/(1 + α)), obtaining, for l sufficiently large, log Z N,ω Np(l)gain/2. Since gain> 0 h < h 1/(1+α) (λ) we are done.

126 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.58/65 On the Regularity of the Free Energy Theorem 8. [G. and Toninelli Alea06] F( ) C in L. The proof is based on estimates on decay of correlations of local observables and on [von Dreifus, Klein and Fernando Perez CMP95] Intriguing issues: Griffiths singularities?

127 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.59/65 Regularity at criticality Stated for pinning (β λ for copolymers) Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of disorders (including ω 1 bounded and ω 1 N(0,1)), for every β > 0 there exists C such that for every h F(β,h) (1 + α)c (h h c (β)) 2.

128 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.59/65 Regularity at criticality Stated for pinning (β λ for copolymers) Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of disorders (including ω 1 bounded and ω 1 N(0,1)), for every β > 0 there exists C such that for every h F(β,h) (1 + α)c (h h c (β)) 2. Possibly more transparent when written as 0 F(β,h) F(β,h c (β)) (1 + α)c (h h c (β)) 2, (the result is non trivial only for h < h c (β).)

129 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.59/65 Regularity at criticality Stated for pinning (β λ for copolymers) Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of disorders (including ω 1 bounded and ω 1 N(0,1)), for every β > 0 there exists C such that for every h F(β,h) (1 + α)c (h h c (β)) 2. Rephrasing once more: F(λ, ) is C 1,1 at h c (β) = the transition is at least of second order (almost third...)

130 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.60/65 Smoothing? Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90]

131 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.60/65 Smoothing? Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90] We need to precise: the pure (i.e. non disordered) model the regularity of F for the pure model

132 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.60/65 Smoothing? Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90] We need to precise: the pure (i.e. non disordered) model the regularity of F for the pure model The pure models are: Pinning: set β = 0 just reward h per visit. Explicitly solvable (seen). Copolymer: periodic (T ) distribution of charges. (Almost) explicitly solvable too.

133 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.61/65 Smoothing! (Sometimes) In all cases: F pure (h) hրh c (h c h) 1/ min(α,1)ˆl(hc h),

134 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.61/65 Smoothing! (Sometimes) In all cases: F pure (h) hրh c (h c h) 1/ min(α,1)ˆl(hc h), Disordered above / pure below Smoothing: like α = 1/2 0 RW model PS model... 1/4 1/3 α = 1/2 α = th 3rd order second order first order

135 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.61/65 Smoothing! (Sometimes) In all cases: F pure (h) hրh c (h c h) 1/ min(α,1)ˆl(hc h), Disordered above / pure below Smoothing: like α = 1/2 0 RW model PS model... 1/4 1/3 α = 1/2 α = th 3rd order second order first order Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2)

136 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.61/65 Smoothing! (Sometimes) In all cases: F pure (h) hրh c (h c h) 1/ min(α,1)ˆl(hc h), Disordered above / pure below Smoothing: like α = 1/2 0 RW model PS model... 1/4 1/3 α = 1/2 α = th 3rd order second order first order Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2) [Forgacs et al. 1986], [Derrida et al. 1992]... (!)

137 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.61/65 Smoothing! (Sometimes) In all cases: F pure (h) hրh c (h c h) 1/ min(α,1)ˆl(hc h), Disordered above / pure below Smoothing: like α = 1/2 0 RW model PS model... 1/4 1/3 α = 1/2 α = th 3rd order second order first order Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2) [Cule, Hwa 97], [Tang,Chaté 01], [Garel, Monthus 05], [Coluzzi 05] (Biomath: DNA denaturation, Poland-Scheraga model)

138 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.62/65 Rare stretch strategy (again) On the proof of: F(λ,h) (1 + α)c(β)(h h c (β)) 2 (GT06) Picture to keep in mind: N and then l S n Atypical ω n s 0 l N n

139 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.63/65 On the proof of the smoothing inequality (Th. 9) Restricted set-up: ω 1 N(0,1) and only pinning model (energy= P n τ [1,N] (βω n h)).

140 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.63/65 On the proof of the smoothing inequality (Th. 9) Restricted set-up: ω 1 N(0,1) and only pinning model (energy= P n τ [1,N] (βω n h)). Note that, observing 1 l l n=1 ω n δ/β implies for l large (roughly) observing ω j N(δ/β,1) for j = 1,...,l and therefore log Z l,ω,β,h lf(β,h δ).

141 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.63/65 On the proof of the smoothing inequality (Th. 9) Restricted set-up: ω 1 N(0,1) and only pinning model (energy= P n τ [1,N] (βω n h)). Note that, observing 1 l l n=1 ω n δ/β implies for l large (roughly) observing ω j N(δ/β,1) for j = 1,...,l and therefore log Z l,ω,β,h lf(β,h δ). Precisely: lim l 1 l log P(log Z l,ω,β,h lf(β,h δ)/2) 1 2 ( ) δ 2. β

142 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.64/65 On the proof of the smoothing inequality (Th. 9) Now apply the rare stretch strategy: ( ( ) 1 δ 2 F(β,h) p(l) F(β,h δ) 2(1 + α) + o l (1)). 2 β

143 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.64/65 On the proof of the smoothing inequality (Th. 9) Now apply the rare stretch strategy: ( ( ) 1 δ 2 F(β,h) p(l) F(β,h δ) 2(1 + α) + o l (1)). 2 β Set h = h c (β), so that the LHS is zero and we obtain that for every l F(β,h δ) 2(1 + α) ( δ β) 2 + o l (1).

144 Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.64/65 On the proof of the smoothing inequality (Th. 9) Now apply the rare stretch strategy: ( ( ) 1 δ 2 F(β,h) p(l) F(β,h δ) 2(1 + α) + o l (1)). 2 β Set h = h c (β), so that the LHS is zero and we obtain that for every l F(β,h δ) 2(1 + α) ( δ β) 2 + o l (1). By letting l we obtain the smoothing inequality. Obs.: no CLT and no boundary! Rather: LDs and flexibility.

145 THE END Stochastic Processes in Mathematical Physics, Firenze June 19 23, 2006 p.65/65