Comportement critique du modèle d accrochage de polymère Niccolò Torri Toulouse, 29 Mars 2016
1 Pinning Model 2 New Results 3 Proof 4 Perspectives N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 1 / 16
Pinning Model Random Walk (S, P), S Z s n n Allow interaction with 0. Region of interaction {1,..., N} Contact process τ = {n : S n = 0} N 0 τ 1 τ 2 τ k N ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω N-2 ω N-1 ω N ω N+1 ω N+2 ω N+3 Interaction Perturb any τ i N by a Exponential Reward exp {βω i + h} Behavior of S perturbed by ω (DISORDER) N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 2 / 16
Pinning Model Random Walk (S, P), S Z s n n Allow interaction with 0. Region of interaction {1,..., N} Contact process τ = {n : S n = 0} N 0 τ 1 τ 2 τ k N ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω N-2 ω N-1 ω N ω N+1 ω N+2 ω N+3 Interaction P ω N, β, h (τ) = 1 Z ω N,β,h exp (βω i + h)1 {i τ} P(τ) i Ω N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 2 / 16
Physical Interpretation Interaction Polymer/membrane s n n (Directed Random Walk) (n, S n ) Directed Polymer DNA-denaturation (Poland-Scheraga Model) τ Contact process Sequence of Loops and bonded bases N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 3 / 16
Assumptions Contact process τ N 0 (τ i τ i 1 ) i N i.i.d. & P(τ i τ i 1 = n) c n 1+α, α > 0 disorder (quenched realisation of) (ω = (ω x ) x Ω, P) i.i.d. E(ω 1 ) = 0, Var(ω 1 ) = 1, Λ(z) = log E(e zω 1 ) <, z small. N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 4 / 16
Assumptions Contact process τ N 0 (τ i τ i 1 ) i N i.i.d. & P(τ i τ i 1 = n) c n 1+α, α > 0 disorder (quenched realisation of) (ω = (ω x ) x Ω, P) i.i.d. E(ω 1 ) = 0, Var(ω 1 ) = 1, Λ(z) = log E(e zω 1 ) <, z small. Goal: Study (τ, PN, ω β, h ) when N gets large. τ [0, N] c h,β N or τ [0, N] = o(n)? Phase Transition? Critical point? N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 4 / 16
Localization/Delocalization Free Energy 1 F(β, h) := lim N N E [ ] log Z ω β,h,n ( ) {points of τ N} F(β, h) = lim h N EEω β,h,n N ( ) Critical Point β 0 there exists a critical point h c (β) h c (β) := sup{h : F(β, h) = 0} s n Delocalized Localized n h > h c (β) localization, ( ) > 0, h < h c (β) de-localization, ( ) = 0. N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 5 / 16
Analysis of the model Goal: understand h c (β). Homogeneous model h c (0) explicit. It provides Lower/Upper bounds h c (0) Λ(β) h c (β) < h c (0), h a c (β) := h c(0) Λ(β) annealed critical point. N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 6 / 16
Relevance/Irrelevance of the disorder α > 1/2 relevant disorder 0 < α < 1/2 irrelevant disorder h c(β) > h c(0)-λ(β) β>0 h c(β) = h c(0)-λ(β) if β small h (β) c h (0) c h (0)-Λ(β) c N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 7 / 16
Relevance/Irrelevance of the disorder α > 1/2 relevant disorder h c(β) > h c(0)-λ(β) β>0 h (β) c h (0) c h (0)-Λ(β) c Aim: for α > 1/2 compute h c (β) hc a (β) as β 0 (Weak Disorder) N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 7 / 16
An overview of the literature case α > 1 Theorem (Q. Berger, F. Caravenna, J. Poisat, R. Sun, N. Zygouras, 2014) Let α > 1, then h c (β) h a c (β) β 0 1 α 2E(τ 1 ) 1 + α β2, case α (1/2, 1) several authors K. S. Alexander, B. Derrida, G. Giacomin, H. Lacoin, F. L. Toninelli and N. Zygouras (2008 2011): Theorem Let α (1/2, 1), then there exist 0 < c < C < such that for β small. c β 2α 2α 1 hc (β) hc a 2α (β) C β 2α 1, N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 8 / 16
1 Pinning Model 2 New Results 3 Proof 4 Perspectives
Results Theorem (Caravenna, Toninelli, T., 2015) Universal feature of the Free Energy: α (1/2, 1) Continuum critical point ( Φ (α) F ˆβ ε ( ˆβ, α 1 2, ĥ ε α) ĥ) = lim ε 0 ε ĥ (α) ( ˆβ) = ĥ (α) (1) ˆβ 2α 2α 1 Theorem (Caravenna, Toninelli, T., 2015) Universal Critical Behavior h c (β) h a c (β) β 0 ĥ (α) (1) β 2α 2α 1, N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 9 / 16
1 Pinning Model 2 New Results 3 Proof 4 Perspectives
Continuum Partition Function Discrete partition [ function { Z ω N,β,h = E exp N n=1 (βω }] n + h)1 n τ Z [exp ω,c β,h (M, { N) = E N n=m+1 (βω } ] n + h)1 n τ M τ, N τ Theorem (Caravenna, Sun, Zygouras, 14) If β N = ˆβN 1 2 α and h N = ĥn α, then ( ( ) Z ω Nt,β Ẑ N,h N )t [0, ) W t, ˆβ,ĥ (d) N ( Z ω,c β N,h N (Ns, Nt) ) (s,t) [0, ) 2 (d) N t [0, ) ( Ẑ W,c ˆβ,ĥ (s, t) ) (s,t) [0, ) 2 N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 10 / 16
Continuum Partition Function, Remarks τ := {n : S n = 0} ετ ε 0 ˆτ regenerative set Continuum ingredients: - regenerative set ˆτ (Cont. contact process) - White Noise (Cont. disorder) No Gibbs representation Wiener Chaos Expansion Ẑ W t, ˆβ,ĥ = 1 + 1 k! k=1 where ˆϕ (k) t (x 1,, x k ) = (k) ˆϕ (x 1,,x k ) [0,t] k t (x 1,, x k ) k [ ] ˆβW(dx i ) + ĥdx i k=1 1 : density of visited points by ˆτ x 1 α (x 1 k x k 1 ) 1 α N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 11 / 16
Proof Compare free energies L = t ε size of the system: t continuum, ε discrete Φ (α) ( ˆβ, ĥ) = lim t lim ε 0 1 t E log Z ω t/ε,β ε,h ε ε 1 F ( ˆβ ε α 1 2, ĥ ε α ) = lim t 1 t E log Z ω t/ε,β ε,h ε GOAL: interchange the limits η > 0 ε 0 : ε < ε 0 Φ (α) ( ˆβ, ĥ η) ε 1 F ( ˆβ ε α 1 2, ĥ ε α) Φ (α) ( ˆβ, ĥ + η) These inequalities imply both our results! N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 12 / 16
Proof - Strategy: Coarse-Graining Compare lim t lim ε & lim ε lim t Coarse-Graining of ετ and ˆτ 0 J 1= 1 J 2= 2 3 J 3= 4 J m= t 5 6 s t 1 s2 t 2 s3 t 3 sm t m t 1 t Partition function factorization (of Z ω t/ε,β ε,h ε and Ẑ W t, ˆβ,ĥ ) Z c 1(τ) Z c 2(τ) Z c 3(τ) Z c 4(τ) Convergence on each block t t Couple convergence of ( si (ε), t i (ε) ) with Z c t/ε( a, b ) Uniform control of the error on any visited block. Conclusion: η > 0 ε 0 : ε < ε 0 C η ε,t : (1) C η ε,t Ẑ W t, ˆβ,ĥ η Z ω t/ε,β ε,h ε C η ε,t Ẑ W t, ˆβ,ĥ+η ) 0 (2) ε < ε 0 : lim sup t 1 t E ( log C η ε,t s i t i (Ԑ) (Ԑ) Ԑ 0 s i t i & c, disc Ԑ 0 Z (τ) i c, cont Z i (τ) N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 13 / 16
Proof - Strategy: Coarse-Graining Compare lim t lim ε & lim ε lim t Coarse-Graining of ετ and ˆτ 0 J 1= 1 J 2= 2 3 J 3= 4 J m= t 5 6 s t 1 s2 t 2 s3 t 3 sm t m t 1 t Partition function factorization (of Z ω t/ε,β ε,h ε and Ẑ W t, ˆβ,ĥ ) Z c 1(τ) Z c 2(τ) Z c 3(τ) Z c 4(τ) Convergence on each block t t Couple convergence of ( si (ε), t i (ε) ) with Z c t/ε( a, b ) Uniform control of the error on any visited block. Conclusion: η > 0 ε 0 : ε < ε 0 C η ε,t : (1) C η ε,t Ẑ W t, ˆβ,ĥ η Z ω t/ε,β ε,h ε C η ε,t Ẑ W t, ˆβ,ĥ+η ) 0 (2) ε < ε 0 : lim sup t 1 t E ( log C η ε,t s i t i (Ԑ) (Ԑ) Ԑ 0 s i t i & c, disc Ԑ 0 Z (τ) i c, cont Z i (τ) Similar to Copolymer Model (den Hollander & Bolthausen, 1997 and Caravenna & Giacomin, 2010) N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 13 / 16
1 Pinning Model 2 New Results 3 Proof 4 Perspectives
Directed Polymer in Random Environment (DPRE) Region of interaction {1,, N} Z (directed) (simple) random Walk (n, S n ) n N Interaction P ω N, β, h (S) = 1 Z ω exp β ω (i,x) 1 {Si =x} N,β P(S) (i,x) Ω N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 14 / 16
Background Disorder assumption: (ω, P) i.i.d. with E(ω 1,1 ) = 0, Var(ω 1,1 ) = 1 and Λ(s) < for s small. Existence of a critical point β c [0, ) : β < β c Diffusivity, β > β c Superdiffusivity. β c connected with the free energy F(β): In dimension d = 1 we have β c = sup{β : F(β) = 0} = 0 Goal: understand the free energy as β 0 N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 15 / 16
Conjectures for DPRE Theorem (Alberts, Khanin, Quastel, 2014) Continuum model: If β = ˆβN 1/4, then ( ( ) Z ω (d) Nt,β N Ẑ )t [0, ) W N t, ˆβ t [0, ) Conjecture: Continuum model describe the free energy asymptotics ( 1 ) Φ DPRE F ε 4 (1) = lim ε 0 ε F ( β ) β 4 Φ DPRE (1) N. Torri (Université de Nantes) Polymères aléatoires Toulouse, 29 Mars 2016 16 / 16
Merci pour votre attention!