Universalité pour des polymères aléatoires. Paris, 17 Novembre 2015

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1 Universalité pour des polymères aléatoires Niccolò Torri Paris, 17 Novembre 2015

2 1 Polymers 2 Pinning Model and DPRE 3 Universality of the Pinning Model New Results 4 Proofs Continuum model Proof of our results N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

3 Polymers Interactions with Itself External environment Depend on some parameters Aim Spatial configuration Phase transition? Critical Points? N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

4 Intermezzo: Some Basic Probabilistic Processes Random Walk on Z d S n = n i=1 X i, X i = increments. x i { Self Avoiding Random Walk (SAW) on Z d Conditioned to visit at most ones each state. N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

5 Abstract Monomers and Abstract Polymers Increment X i a monomer Use SAW: Abstract Polymer (N monomers): N-increments of a SAW N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

6 Abstract Monomers and Abstract Polymers Increment X i a monomer Use SAW: SAW: challenging object! Subclass: Directed Random Walks (Directed Polymers) Abstract Polymer (N monomers): N-increments of a SAW s n n N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

7 1 Polymers 2 Pinning Model and DPRE 3 Universality of the Pinning Model New Results 4 Proofs Continuum model Proof of our results

8 Interaction with the environment Pinning Model interaction with a membrane Directed Polymer in Random Environment (DPRE) interaction with the whole environment s n n region of interaction : straight line N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

9 Pinning Model s n n region of interaction : straight line Region of interaction Lattice Ω = {1,, N} Contact process Renewal Process τ = {n : S n = 0} N 0 Interaction P(τ i τ i 1 = n) P ω N, β, h (τ) = 1 Z ω N,β,h c n 1+α, α > 0 exp (βω i + h)1 {i τ} P(τ) i Ω N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

10 Directed Polymer in Random Environment (DPRE) Region of interaction Lattice Ω = {1,, N} Z sn =X Xn Polymer Directed Walk (n, S n ) n N Ω Interaction P ω N, β, h (S) = 1 Z ω exp (βω (i,x) + h)1 {Si =x} N,β,h P(S) (i,x) Ω x p(x,x+1) = 1 2 p(x,x - 1) = 1 2 n N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

11 1 Polymers 2 Pinning Model and DPRE 3 Universality of the Pinning Model New Results 4 Proofs Continuum model Proof of our results

12 Behavior of the model Pinning Model: interaction polymer and membrane. Renewal 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 Pinning Model τ [0, N] perturbed P ω N, β, h (τ) = 1 N Z ω exp (βω i + h)1 {i τ} P(τ) N,β,h 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. i=1 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

13 Behavior of the model Pinning Model: interaction polymer and membrane. Renewal 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 Pinning Model τ [0, N] perturbed P ω N, β, h (τ) = 1 N Z ω exp (βω i + h)1 {i τ} P(τ) N,β,h 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. i=1 Goal: Study τ 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) Random Polymers Paris, 17 Novembre / 26

14 Localization/Delocalization Free Energy F (α) 1 (β, h) = lim N N E [ ] log Z ω β,h,n ( ) {points of τ N} h F (α) (β, h) = lim EE ω N β,h,n N ( ) β 0 there exists a critical point h c (β) s n Delocalized Localized n h > h c (β) localization, ( ) > 0, h < h c (β) de-localization, ( ) = 0. N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

15 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 c (0) Λ(β) annealed critical point. N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

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 c (0) Λ(β) annealed critical point. P ω N, β, h (τ) = 1 Z ω N,β,h N exp (βω i + h)1 {i τ} P(τ) i=1 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

17 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) Random Polymers Paris, 17 Novembre / 26

18 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: When α > 1/2, asymptotics of h c (β) as β 0 (Weak Disorder). N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

19 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) β 0 cβ 2, where c is explicit depending on the law of τ i τ i 1. case α (1/2, 1) several authors K. S. Alexander, B. Derrida, G. Giacomin, H. Lacoin, F. L. Toninelli and N. Zygouras ( ): Theorem Let α (1/2, 1), then there exist 0 < c < C < such that for β small. c β 2α 2α 1 hc (β) C β 2α 2α 1, N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

20 3 Universality of the Pinning Model New Results

21 Results Theorem (Caravenna, Toninelli, T., 2015) Universal feature of the Free Energy: α (1/2, 1) F (α) F ( ˆβ, (α)( ˆβ ε α 1 2, ĥ ε α) ĥ) = lim ε 0 ε Rescale relation F (α) ( ˆβ, ĥ) = β ( ) 2 2α 1 F (α) 1, ĥβ 2α 2α 1 ĉ universal constant depending on α and c. H (α) ( ˆβ) = ĉ ˆβ 2α 2α 1 Theorem (Caravenna, Toninelli, T., 2015) Universal Critical Behavior h c (β) β 0 ĉ β 2α 2α 1, N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

22 Conjectures for DPRE Universal feature of the Free Energy F DPRE F (α)( ε (1) = lim ε 0 ε 1 ) 4 F (α)( ε ) ε 4 F DPRE (1). Conjecture supported by Lacoin (2009) (no sharp result) N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

23 4 Proofs Continuum model Proof of our results

24 General Framework Discrete Lattice Ω = Ω Z d, Reference System (σ = (σ x ) x Ω, P Ω ), σ x {0, 1}, 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. Interaction (disordered measure) P ω Ω,β,h (dσ) = 1 Z ω exp (βω i + h)σ x P Ω(dσ) N,β,h x Ω N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

25 General Framework Discrete Lattice Ω = Ω Z d, Reference System (σ = (σ x ) x Ω, P Ω ), σ x {0, 1}, Interaction (disordered measure) P ω Ω,β,h (dσ) = 1 Z ω exp (βω i + h)σ x P Ω(dσ) N,β,h x Ω Pinning Model DPRE Lattice Ω = (0, N) Ω = (0, N) R Reference System σ x = 1 {x τ} σ (n,x) = 1 {Sn =x} N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

26 General Framework: Goal Pinning Model DPRE Lattice Ω = (0, N) Ω = (0, N) R Reference System σ x = 1 {x τ} σ (n,x) = 1 {Sn =x} Assumption: Continuum Limit rescaling δ = δ N of Ω s.t. (σ = (σ x ) x Ωδ, P Ωδ ) δ 0 ( ˆσ = ( ˆσ x ) x Ω, ˆP Ω ) Pinning Model DPRE Rescaling δ = 1 N δ = ( 1 N, 1 ) N 1/2 Continuum limit Regenerative set ˆτ Brownian Motion (B t ) t Continuum system ˆσ x = 1 {x ˆτ} ˆσ x = 1 {Bt =y}, N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

27 General Framework: Goal Convergence of reference model: (σ = (σ x ) x Ωδ, P Ωδ ) δ 0 ( ˆσ = ( ˆσ x ) x Ω, ˆP Ω ) Convergence of the disordered model: rescaling of β, h = β, h(δ) δ 0 0: (σ = (σ x ) x Ωδ, P ω Ω δ,β δ,h δ ) δ 0? Idea look at the partition function Z ω Ω,β,h...Because it defines completely P ω Ω,β,h N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

28 Chaos Expansion Partition function Z ω Ω,β,h = E [ ] Ω e n Ω(βω n +h)σ n Polynomial Chaos Expansion where Z ω Ω,β,h = E Ω (1 + ε n σ n ) = n Ω Ω ϕ (k) k! Ω (n 1,, n k ) k=1 (n 1,,n k ) Ω k ε n = e (βω n+h) 1 ϕ (k) Ω (n 1,, n k ) = E Ω [ σn1 σ nk ] k k=1 ε ni N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

29 Continuum Model Theorem (Caravenna, Sun, Zygouras (2014)) Assumption γ : (δ γ ) k ϕ (k) Ω δ (x 1,, x k ) δ 0 ˆϕ (k) Ω (x 1,, x k ). Choosing β δ = ˆβδ d/2 γ, h δ = ĥδ d γ Z ω Ω δ,β δ,h δ (d) δ 0 ẐW Ω, ˆβ,ĥ Continuum Partition function: Wiener Chaos Expansion Ẑ W Ω, ˆβ,ĥ = k! k=1 (k) ˆϕ (x 1,,x k ) Ω k Ω (x 1,, x k ) k [ ] ˆβW(dx i ) + ĥdx i k=1 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

30 Proof Ω δ Z ω Ω δ,β,h = 1 + k=1 1 k! (x 1,,x k ) Ω k δ Idea Lindeberg Principle: ε x is quite Gaussian : ϕ (k) Ω δ (x 1,, x k ) k k=1 ε xi ε x = e βω x+h 1 : Var(ε x ) β 2, E(ε x ) h + β2 2 = h : ε x βw 1 + h = β x d/2 W x + h (x 1,,x k ) Ω k δ ϕ (k) Ω δ (x 1,, x k ) (k) ϕ (x 1,,x k ) Ω k k k=1 ε xi Ω δ (x 1,, x k ) k [ ] βδ d/2 W(dx i ) + h δ d dx i k=1 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

31 Example: Pinning Model Polynomial Chaos expansion: ϕ (k) Ω (n [ ] [ ] 1,, n k ) = E Ω σn1 σ nk = EΩ 1n1 τ 1 nk τ Theorem (Doney (1997)) P(τ 1 = n) c, α (0, 1) P(n τ) c n 1+α n 1 α N Z ω Ω,β,h 1 + k=1 0<n 1 < <n k <N C k n 1 α 1 (n k n k 1 ) 1 α k k=1 ε xi N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

32 Example: Pinning Model N Z ω Ω,β,h 1 + k=1 0<n 1 < <n k <N C k n 1 α 1 (n k n k 1 ) 1 α k k=1 ε xi k=1, contribution of one point + k=2, contribution of two points k=m, contribution of m-points N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

33 Example: Pinning Model N Z ω Ω,β,h 1 + k=1 0<n 1 < <n k <N C k n 1 α 1 (n k n k 1 ) 1 α k k=1 ε xi Rescaling δ = 1 N and Gaussian Approximation Z ω Ω δ,β,h δ k=1 0<x 1 < <x k <1 δ (1 α)k C k x 1 α 1 (x k x k 1 ) 1 α k [ ] βδ 1 2 W(dxi ) + h δ 1 dx i k=1 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

34 Pinning Model and DPRE rescaling Pinning Model: γ = 1 α β δ = ˆβ δ α 1/2, h δ = ĥ δα must be α > 1/2 (Disorder relevance) ˆϕ (k) Ω (x 1,, x k ) = β δ = ˆβ δ 1/4, h δ = ĥ δ C k x 1 α (x 1 k x k 1 ) 1 α DPRE: γ = 2 ˆϕ (k) Ω ((x 1, t 1 ),, (x k, t k ))) = k e (x i x i 1 )2 /2(t i t i 1 ) i=1 2π(ti t i 1 ) N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

35 Universality of the free energy The continuum model captures the critical behavior of the system in the weak disorder regime (β, h 0): Naive approach ] 1 F ( ˆβ, ĥ) := lim [log Ω R d Ω E Ẑ W Ω =limδ 0 δ d Ω δ = Ω, ˆβ,ĥ 1 lim lim Ω R d δ 0 δ d Ω δ E [ ] log Z ω!!! Ω δ,β δ,h δ = lim lim 1 δ 0 Ω R d δ d Ω δ E [ ] log Z ω F(β δ, h δ ) Ω δ,β δ,h δ = lim. δ 0 δ d N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

36 Universality of the free energy The continuum model captures the critical behavior of the system in the weak disorder regime (β, h 0): Naive approach ] 1 F ( ˆβ, ĥ) := lim [log Ω R d Ω E Ẑ W Ω =limδ 0 δ d Ω δ = Ω, ˆβ,ĥ 1 lim lim Ω R d δ 0 δ d Ω δ E [ ] log Z ω!!! Ω δ,β δ,h δ = lim lim 1 δ 0 Ω R d δ d Ω δ E [ ] log Z ω F(β δ, h δ ) Ω δ,β δ,h δ = lim. δ 0 δ d Universal asymptotics F(β δ, h δ ) δ d ˆF(1, 1) as δ 0 N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

37 4 Proofs Continuum model Proof of our results

38 Recall the Results: Universal Critical Behavior Theorem (Caravenna, Toninelli, T., 2015) Universality of the Free Energy: F (α) F ( ˆβ, (α)( ˆβ ε α 1 2, ĥ ε α) ĥ) = lim ε 0 ε Theorem (Caravenna, Toninelli, T., 2015) h c (β) β 0 ĉ β 2α 2α 1, where ĉ universal constant depending on α and c. N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

39 Proof - Continuum Model τ := {n : S n = 0} ετ ε 0 Thm (C.S.Z.,15) (cond.) Pinning Model converges α (1/2, 1), β = ˆβε α 1 2, h = ĥε α : t 1 t 2 ετ β,h (d) ε 0 ˆτ ˆτ ˆβ,ĥ Continuum ingredients: - regenerative set (ˆτ) - White Noise (Cont. disorder) Problem: No Gibbs representation Wiener Chaos Expansion N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

40 Proof - Strategy: Coarse-Graining N = t ε. Consider 1 N E log Z N Convergence on each block 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 decomposition t s i t i (Ԑ) (Ԑ) Ԑ 0 s i t i Technical difficulty & c, disc Ԑ 0 Z (τ) Couple together convergence of ( si (ε), t i (ε) ) with Z c t/ε( a, b ) i c, cont Z i (τ) Z c 1(τ) Z c 2(τ) Z c 3(τ) Z c 4(τ) Z c,disc. i Z c,cont. i ( (τ/ε) = Z c t/ε si (ε), ) t i (ε) ( ) (ˆτ) = Z c si, t i t t Get: η > 0 ε 0 : ε < ε 0 F (α) ( ˆβ, ĥ η) ε 1 F (α)( ˆβ ε α 1 2, ĥ ε α) F (α) ( ˆβ, ĥ + η) N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

41 Proof - Strategy: Coarse-Graining N = t ε. Consider 1 N E log Z N Convergence on each block 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 decomposition t s i t i (Ԑ) (Ԑ) Ԑ 0 s i t i Technical difficulty & c, disc Ԑ 0 Z (τ) Couple together convergence of ( si (ε), t i (ε) ) with Z c t/ε( a, b ) i c, cont Z i (τ) Z c 1(τ) Z c 2(τ) Z c 3(τ) Z c 4(τ) Z c,disc. i Z c,cont. i ( (τ/ε) = Z c t/ε si (ε), ) t i (ε) ( ) (ˆτ) = Z c si, t i t t Get: η > 0 ε 0 : ε < ε 0 F (α) ( ˆβ, ĥ η) ε 1 F (α)( ˆβ ε α 1 2, ĥ ε α) F (α) ( ˆβ, ĥ + η) Similar to Copolymer Model (den Hollander & Bolthausen, 1997 and Caravenna & Giacomin, 2010) N. Torri (Université de Nantes) Random Polymers Paris, 17 Novembre / 26

42 Merci!

43 Merci!