The Models Free Energy Path Results Proof Wetting and pinning models The choice of H N ( ϕ ) We rather consider the Laplacian case: N � � � H N ( ϕ ) := V ∆ ϕ i i =0 ∆ ϕ i := ∇ ϕ i +1 − ∇ ϕ i = ϕ i +1 + ϕ i − 1 − 2 ϕ i ◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34
The Models Free Energy Path Results Proof Wetting and pinning models The choice of H N ( ϕ ) We rather consider the Laplacian case: N � � � � � + � H N ( ϕ ) := V ∆ ϕ i V ∇ ϕ i i =0 ∆ ϕ i := ∇ ϕ i +1 − ∇ ϕ i = ϕ i +1 + ϕ i − 1 − 2 ϕ i ◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations ◮ ∇ and ∆ together? Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34
The Models Free Energy Path Results Proof Wetting and pinning models The choice of H N ( ϕ ) We rather consider the Laplacian case: N � � � H N ( ϕ ) := V ∆ ϕ i i =0 ∆ ϕ i := ∇ ϕ i +1 − ∇ ϕ i = ϕ i +1 + ϕ i − 1 − 2 ϕ i ◮ Simplest non-nearest neighbor interaction ◮ Semi-flexible polymers: ∆ favors affine configurations ◮ ∇ and ∆ together? Interpretation of the free case ε = 0: ◮ P p 0 , N is (the bridge of) the integral of a random walk ◮ P w 0 , N is further conditioned to stay ≥ 0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 11 / 34
The Models Free Energy Path Results Proof Wetting and pinning models Laplacian interaction in ( d + 1)-dimension Fields ϕ : { 1 , . . . , N } d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34
The Models Free Energy Path Results Proof Wetting and pinning models Laplacian interaction in ( d + 1)-dimension Fields ϕ : { 1 , . . . , N } d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case: [Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07] Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34
The Models Free Energy Path Results Proof Wetting and pinning models Laplacian interaction in ( d + 1)-dimension Fields ϕ : { 1 , . . . , N } d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case: [Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07] Henceforth we study P p ε, N and P w ε, N with Laplacian interaction and boundary conditions ϕ − 1 = ϕ 0 = ϕ N = ϕ N +1 = 0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34
The Models Free Energy Path Results Proof Wetting and pinning models Laplacian interaction in ( d + 1)-dimension Fields ϕ : { 1 , . . . , N } d → R with Laplacian interaction for d ≥ 2 are models for semiflexible membranes The problem of entropic repulsion (probability for the field to stay non-negative) has been studied in the Gaussian case: [Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07] Henceforth we study P p ε, N and P w ε, N with Laplacian interaction and boundary conditions ϕ − 1 = ϕ 0 = ϕ N = ϕ N +1 = 0 Assumptions on V : � � � e − V ( x ) d x = 1 , x e − V ( x ) d x = 0 , x 2 e − V ( x ) d x = 1 R R R + regularity: x �→ e − V ( x ) continuous and V (0) < + ∞ . Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 12 / 34
The Models Free Energy Path Results Proof Outline 1. The Models Introduction Wetting and pinning models 2. Free Energy Results The free energy The phase transition The disordered case 3. Path Results Path results Refined critical scaling limit 4. Sketch of the Proof Integrated random walk Markov renewal theory Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 13 / 34
The Models Free Energy Path Results Proof The free energy How to define localization and delocalization? Recall the partition function: (zero boundary conditions) � N − 1 � � � Z a e −H N ( ϕ ) ε, N = d ϕ i + ε δ 0 ( d ϕ i ) Ω a i =1 N N = R N − 1 while Ω w where a ∈ { p , w } and Ω p N = [0 , ∞ ) N − 1 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34
The Models Free Energy Path Results Proof The free energy How to define localization and delocalization? Recall the partition function: (zero boundary conditions) � N − 1 � � � Z a e −H N ( ϕ ) ε, N = d ϕ i + ε δ 0 ( d ϕ i ) Ω a i =1 N N = R N − 1 while Ω w where a ∈ { p , w } and Ω p N = [0 , ∞ ) N − 1 Free Energy 1 F a ( ε ) := N log Z a lim (super-additivity) ε, N N →∞ Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34
The Models Free Energy Path Results Proof The free energy How to define localization and delocalization? Recall the partition function: (zero boundary conditions) � N − 1 � � � Z a e −H N ( ϕ ) ε, N = d ϕ i + ε δ 0 ( d ϕ i ) Ω a i =1 N N = R N − 1 while Ω w where a ∈ { p , w } and Ω p N = [0 , ∞ ) N − 1 Free Energy 1 F a ( ε ) := N log Z a lim (super-additivity) ε, N N →∞ Basic observation: F a ( ε ) ≥ F a (0) = 0 for all ε ≥ 0 and a ∈ { p , w } Z a ε, N ≥ Z a 0 , N ≈ N − c ( c > 0) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 14 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: c then ℓ N N → D a ( ε ) > 0 in P a ◮ if ε > ε a ε, N –probability Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: c then ℓ N N → D a ( ε ) > 0 in P a ◮ if ε > ε a ε, N –probability c then ℓ N ◮ if ε < ε a N → 0 in P a ε, N –probability (delocalization) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: c then ℓ N N → D a ( ε ) > 0 in P a ◮ if ε > ε a ε, N –probability c then ℓ N ◮ if ε < ε a N → 0 in P a ε, N –probability (delocalization) ◮ if ε = ε a c ? Depends on the model: Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: c then ℓ N N → D a ( ε ) > 0 in P a ◮ if ε > ε a ε, N –probability c then ℓ N ◮ if ε < ε a N → 0 in P a ε, N –probability (delocalization) ◮ if ε = ε a c ? Depends on the model: c + h ) = o ( h ) [ > 1 st order trans.] ε = ε a ◮ if F a ( ε a c is delocalized Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The free energy Localization and delocalization Definition The a -model at ε ≥ 0 is said to be localized if F a ( ε ) > 0. ε > ε a c := sup { ε ≥ 0 : F a ( ε ) = 0 } ∈ [0 , ∞ ] ⇐ ⇒ localized � � Setting ℓ N := # i ≤ N : ϕ i = 0 we have: c then ℓ N N → D a ( ε ) > 0 in P a ◮ if ε > ε a ε, N –probability c then ℓ N ◮ if ε < ε a N → 0 in P a ε, N –probability (delocalization) ◮ if ε = ε a c ? Depends on the model: c + h ) = o ( h ) [ > 1 st order trans.] ε = ε a ◮ if F a ( ε a c is delocalized c + h ) ≥ C h [1 st order trans.] ε = ε a ◮ if F a ( ε a c may be localized (phase coexistence, dependence of boundary conditions) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 15 / 34
The Models Free Energy Path Results Proof The phase transition The phase transition Theorem ([CD1]) Both P p ε, N and P w ε, N undergo a non-trivial phase transition: 0 < ε p c < ε w < ∞ c and F a ( ε ) is analytic on [0 , ε a c ) ∪ ( ε a c , ∞ ) . (variational formula) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34
The Models Free Energy Path Results Proof The phase transition The phase transition Theorem ([CD1]) Both P p ε, N and P w ε, N undergo a non-trivial phase transition: 0 < ε p c < ε w < ∞ c and F a ( ε ) is analytic on [0 , ε a c ) ∪ ( ε a c , ∞ ) . (variational formula) ◮ In the pinning model the transition is exactly of 2 nd order: h ≤ F p ( ε p C 1 c + h ) ≤ o ( h ) log 1 h Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34
The Models Free Energy Path Results Proof The phase transition The phase transition Theorem ([CD1]) Both P p ε, N and P w ε, N undergo a non-trivial phase transition: 0 < ε p c < ε w < ∞ c and F a ( ε ) is analytic on [0 , ε a c ) ∪ ( ε a c , ∞ ) . (variational formula) ◮ In the pinning model the transition is exactly of 2 nd order: h ≤ F p ( ε p C 1 c + h ) ≤ o ( h ) log 1 h ◮ In the wetting model the transition is of 1 st order: � � F w ( ε w c + h ) ∼ C 2 h ℓ N ∼ D N , D > 0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 16 / 34
The Models Free Energy Path Results Proof The phase transition The gradient case Differences in the gradient case ◮ the transition is non-trivial only in the wetting model: ε p , ∇ 0 < ε w , ∇ = 0 , < ∞ c c Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34
The Models Free Energy Path Results Proof The phase transition The gradient case Differences in the gradient case ◮ the transition is non-trivial only in the wetting model: ε p , ∇ 0 < ε w , ∇ = 0 , < ∞ c c ◮ the transition is of 2 nd order: F p � � F w � � ∼ C p h 2 , ε p , ∇ ε w , ∇ ∼ C w h 2 + h + h c c Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34
The Models Free Energy Path Results Proof The phase transition The gradient case Differences in the gradient case ◮ the transition is non-trivial only in the wetting model: ε p , ∇ 0 < ε w , ∇ = 0 , < ∞ c c ◮ the transition is of 2 nd order: F p � � F w � � ∼ C p h 2 , ε p , ∇ ε w , ∇ ∼ C w h 2 + h + h c c F p ( ε ) F w ( ε ) ∇ ∇ skip ∆ ∆ ε p ε w , ∇ ε w ε p , ∇ ε ε c c c c Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 17 / 34
The Models Free Energy Path Results Proof The disordered case A look at the disordered case Disordered version of our model: ( d ϕ p i = d ϕ i and d ϕ w i = d ϕ + i ) N � � � := e −H N ( ϕ ) � � i + ε e βω i δ 0 ( d ϕ i ) P a d ϕ a d ϕ 1 , . . . , d ϕ N ε,β, ω, N Z a ε,β,ω, N i =1 where β ≥ 0 and { ω i } i ∈ N are IID N (0 , 1) (law P indep. P a ). Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34
The Models Free Energy Path Results Proof The disordered case A look at the disordered case Disordered version of our model: ( d ϕ p i = d ϕ i and d ϕ w i = d ϕ + i ) N � � � := e −H N ( ϕ ) � � i + ε e βω i δ 0 ( d ϕ i ) P a d ϕ a d ϕ 1 , . . . , d ϕ N ε,β,ω, N Z a ε,β,ω, N i =1 where β ≥ 0 and { ω i } i ∈ N are IID N (0 , 1) (law P indep. P a ). Quenched free energy F a � � 1 N log Z a ε,β,ω, N ≥ 0 ε, β := lim N →∞ exists P ( d ω )–a.s. and does not depend on ω (self-averaging) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34
The Models Free Energy Path Results Proof The disordered case A look at the disordered case Disordered version of our model: ( d ϕ p i = d ϕ i and d ϕ w i = d ϕ + i ) N � � � := e −H N ( ϕ ) � � i + ε e βω i δ 0 ( d ϕ i ) P a d ϕ a d ϕ 1 , . . . , d ϕ N ε,β,ω, N Z a ε,β,ω, N i =1 where β ≥ 0 and { ω i } i ∈ N are IID N (0 , 1) (law P indep. P a ). Quenched free energy F a � � 1 N log Z a ε,β,ω, N ≥ 0 ε, β := lim N →∞ exists P ( d ω )–a.s. and does not depend on ω (self-averaging) Localization: F a ( ε, β ) > 0 ε > ε a ⇐ ⇒ c ( β ) (critical line) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 18 / 34
The Models Free Energy Path Results Proof The disordered case Smoothing effect of disorder What is the behavior of ε a c ( β ) for small β ? ( ε a c = ε a c (0)) What is the regularity of the transition in the disordered case? Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34
The Models Free Energy Path Results Proof The disordered case Smoothing effect of disorder What is the behavior of ε a c ( β ) for small β ? ( ε a c = ε a c (0)) What is the regularity of the transition in the disordered case? Theorem ([Giacomin and Toninelli, CMP 06]) Both in the ∇ and ∆ case, both for a = p and for a = w : for every β > 0 there exists C β > 0 such that F a � � ≤ C β h 2 ε a c ( β ) + h When disorder is present the transition is at least of 2 nd order Sharp contrast with homogeneous case cf. Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34
The Models Free Energy Path Results Proof The disordered case Smoothing effect of disorder What is the behavior of ε a c ( β ) for small β ? ( ε a c = ε a c (0)) What is the regularity of the transition in the disordered case? Theorem ([Giacomin and Toninelli, CMP 06]) Both in the ∇ and ∆ case, both for a = p and for a = w : for every β > 0 there exists C β > 0 such that F a � � ≤ C β h 2 ε a c ( β ) + h When disorder is present the transition is at least of 2 nd order Sharp contrast with homogeneous case cf. Very general proof: rare-stretches in ω (Large Deviations) proof Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 19 / 34
The Models Free Energy Path Results Proof Outline 1. The Models Introduction Wetting and pinning models 2. Free Energy Results The free energy The phase transition The disordered case 3. Path Results Path results Refined critical scaling limit 4. Sketch of the Proof Integrated random walk Markov renewal theory Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 20 / 34
The Models Free Energy Path Results Proof Some deeper questions We have established the existence of a phase transition: � if ε < ε a o ( N ) � � c ℓ N = ℓ N := # i ≤ N : ϕ i = 0 if ε > ε a ∼ D · N c Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34
The Models Free Energy Path Results Proof Some deeper questions We have established the existence of a phase transition: � if ε < ε a o ( N ) � � c ℓ N = ℓ N := # i ≤ N : ϕ i = 0 if ε > ε a ∼ D · N c Can we say something more precise on ℓ N ? And on the maximum � � � ϕ i � M N := max 1 ≤ i ≤ N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34
The Models Free Energy Path Results Proof Some deeper questions We have established the existence of a phase transition: � if ε < ε a o ( N ) � � c ℓ N = ℓ N := # i ≤ N : ϕ i = 0 if ε > ε a ∼ D · N c Can we say something more precise on ℓ N ? And on the maximum � � � ϕ i � M N := max 1 ≤ i ≤ N Yes in the pinning case and under additional assumptions on V ( · ): ◮ symmetry: V ( x ) = V ( − x ) for every x ∈ R ◮ uniform strict convexity: ∃ γ > 0 s. t. V ( x ) − γ x 2 2 is convex ◮ regularity: x �→ e − V ( x ) is continuous and V (0) < ∞ � � e − V ( x ) d x = 1 x 2 e − V ( x ) d x = 1 R R Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 21 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N � 1 � K N 3 / 2 ≤ M N ≤ K N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε, N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N � 1 � K N 3 / 2 ≤ M N ≤ K N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε, N � (log N ) 2 � ◮ localized regime ε > ε p c : ℓ N ∼ D N and M N = O : Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N � 1 � K N 3 / 2 ≤ M N ≤ K N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε, N � (log N ) 2 � ◮ localized regime ε > ε p c : ℓ N ∼ D N and M N = O : � M N ≤ K (log N ) 2 � N →∞ P p K →∞ lim inf lim = 1 ε, N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N � 1 � K N 3 / 2 ≤ M N ≤ K N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε, N � (log N ) 2 � ◮ localized regime ε > ε p c : ℓ N ∼ D N and M N = O : � M N ≤ K (log N ) 2 � N →∞ P p K →∞ lim inf lim = 1 ε, N ◮ critical regime ε = ε p N 3 / 2 N c : ℓ N ≈ log N and M N ≈ (log N ) c : Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results A closer look at the typical paths [CD2] c : ℓ N = O (1) and M N ≈ N 3 / 2 : ◮ delocalized regime ε < ε p � � N →∞ P p K →∞ lim inf lim ϕ i � = 0 for i ∈ { K , N − K } = 1 ε, N � 1 � K N 3 / 2 ≤ M N ≤ K N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε, N � (log N ) 2 � ◮ localized regime ε > ε p c : ℓ N ∼ D N and M N = O : � M N ≤ K (log N ) 2 � N →∞ P p K →∞ lim inf lim = 1 ε, N ◮ critical regime ε = ε p N 3 / 2 N c : ℓ N ≈ log N and M N ≈ (log N ) c : � 1 � (log N ) 3 / 2 ≤ M N ≤ K N 3 / 2 N 3 / 2 N →∞ P p K →∞ lim inf lim = 1 ε p c , N K log N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 22 / 34
The Models Free Energy Path Results Proof Path results Scaling Limits We rescale and interpolate linearly the field: for t ∈ [0 , 1] N 3 / 2 + ( Nt − ⌊ Nt ⌋ ) ϕ ⌊ Nt ⌋ +1 − ϕ ⌊ Nt ⌋ ϕ N ( t ) := ϕ ⌊ Nt ⌋ � N 3 / 2 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34
The Models Free Energy Path Results Proof Path results Scaling Limits We rescale and interpolate linearly the field: for t ∈ [0 , 1] N 3 / 2 + ( Nt − ⌊ Nt ⌋ ) ϕ ⌊ Nt ⌋ +1 − ϕ ⌊ Nt ⌋ ϕ N ( t ) := ϕ ⌊ Nt ⌋ � N 3 / 2 � t Let { B t } t ∈ [0 , 1] standard BM, I t := 0 B s d s integrated BM �� � Bridge I t t ∈ [0 , 1] := { I t } t ∈ [0 , 1] conditionally on ( B 1 , I 1 ) = (0 , 0) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34
The Models Free Energy Path Results Proof Path results Scaling Limits We rescale and interpolate linearly the field: for t ∈ [0 , 1] N 3 / 2 + ( Nt − ⌊ Nt ⌋ ) ϕ ⌊ Nt ⌋ +1 − ϕ ⌊ Nt ⌋ ϕ N ( t ) := ϕ ⌊ Nt ⌋ � N 3 / 2 � t Let { B t } t ∈ [0 , 1] standard BM, I t := 0 B s d s integrated BM �� � Bridge I t t ∈ [0 , 1] := { I t } t ∈ [0 , 1] conditionally on ( B 1 , I 1 ) = (0 , 0) Theorem (Scaling Limits [CD2]) ϕ N ( t ) } t ∈ [0 , 1] under P p The rescaled field { � ε, N converges in distribution on C ([0 , 1]) as N → ∞ , for every ε ≥ 0 . The limit is ◮ If ε < ε p c , the law of { � I t } t ∈ [0 , 1] ◮ If ε = ε p c or ε > ε p c , the law concentrated on f ( t ) ≡ 0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 23 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime For ε = ε p N 3 / 2 c the field has very large fluctuations ( ≈ (log N ) c ). � � N 3 / 2 ϕ i � � N O O (log N ) 3 / 2 log N 0 N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime For ε = ε p N 3 / 2 c the field has very large fluctuations ( ≈ (log N ) c ). � � N 3 / 2 ϕ i � � N O O (log N ) 3 / 2 log N 0 N Can we extract a non-trivial scaling limit? Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime For ε = ε p N 3 / 2 c the field has very large fluctuations ( ≈ (log N ) c ). � � N 3 / 2 ϕ i � � N O O (log N ) 3 / 2 log N 0 N Can we extract a non-trivial scaling limit? Not in C ([0 , 1]) or D ([0 , 1]): � � 1 the set i ∈ { 1 , . . . , N } : ϕ i = 0 becomes dense in [0 , 1] N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime For ε = ε p N 3 / 2 c the field has very large fluctuations ( ≈ (log N ) c ). � � N 3 / 2 ϕ i � � N O O (log N ) 3 / 2 log N 0 N Can we extract a non-trivial scaling limit? Not in C ([0 , 1]) or D ([0 , 1]): � � 1 the set i ∈ { 1 , . . . , N } : ϕ i = 0 becomes dense in [0 , 1] N Alternative idea: look at the field in a distributional sense Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 24 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime Introduce the random measure (finite, signed) on [0 , 1] � � := (log N ) 5 / 2 µ N d t ϕ ⌊ Nt ⌋ d t = � ϕ N ( t ) d t N 3 / 2 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime Introduce the random measure (finite, signed) on [0 , 1] � � := (log N ) 5 / 2 µ N d t ϕ ⌊ Nt ⌋ d t = � ϕ N ( t ) d t N 3 / 2 evy process of index 2 Let { L t } t ∈ [0 , 1] be the stable symmetric L´ 5 Π( d x ) = c | x | − 7 / 5 d x 0 drift 0 Brownian component Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime Introduce the random measure (finite, signed) on [0 , 1] � � := (log N ) 5 / 2 µ N d t ϕ ⌊ Nt ⌋ d t = � ϕ N ( t ) d t N 3 / 2 evy process of index 2 Let { L t } t ∈ [0 , 1] be the stable symmetric L´ 5 Π( d x ) = c | x | − 7 / 5 d x 0 drift 0 Brownian component The paths of L are a.s. of bounded variation, hence we set � � := L b − L a d L ( a , b ] random measure on [0 , 1] Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime Introduce the random measure (finite, signed) on [0 , 1] � � := (log N ) 5 / 2 µ N d t ϕ ⌊ Nt ⌋ d t = � ϕ N ( t ) d t N 3 / 2 evy process of index 2 Let { L t } t ∈ [0 , 1] be the stable symmetric L´ 5 Π( d x ) = c | x | − 7 / 5 d x 0 drift 0 Brownian component The paths of L are a.s. of bounded variation, hence we set � � := L b − L a d L ( a , b ] random measure on [0 , 1] Theorem ([CD2]) µ N under P p c , N converges in distribution as N → ∞ toward d L . ε p Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 25 / 34
The Models Free Energy Path Results Proof Refined critical scaling limit The critical regime ϕ N ( t ) � � � � � O log N 1 O log N 0 1 d L 0 1 disorder Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 26 / 34
The Models Free Energy Path Results Proof Outline 1. The Models Introduction Wetting and pinning models 2. Free Energy Results The free energy The phase transition The disordered case 3. Path Results Path results Refined critical scaling limit 4. Sketch of the Proof Integrated random walk Markov renewal theory Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 27 / 34
The Models Free Energy Path Results Proof Integrated random walk A random walk viewpoint ( ε = 0) Let { X i } i ∈ N be IID random variables with law P ( X i ∈ d x ) := e − V ( x ) d x E ( X i ) = 0 Var ( X i ) = 1 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34
The Models Free Energy Path Results Proof Integrated random walk A random walk viewpoint ( ε = 0) Let { X i } i ∈ N be IID random variables with law P ( X i ∈ d x ) := e − V ( x ) d x E ( X i ) = 0 Var ( X i ) = 1 Random walk: Y n := X 1 + . . . + X n Integrated random walk: entropic repulsion Z n := Y 1 + . . . + Y n = n X 1 + ( n − 1) X 2 + . . . + X n Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34
The Models Free Energy Path Results Proof Integrated random walk A random walk viewpoint ( ε = 0) Let { X i } i ∈ N be IID random variables with law P ( X i ∈ d x ) := e − V ( x ) d x E ( X i ) = 0 Var ( X i ) = 1 Random walk: Y n := X 1 + . . . + X n Integrated random walk: entropic repulsion Z n := Y 1 + . . . + Y n = n X 1 + ( n − 1) X 2 + . . . + X n The free case ε = 0 ◮ The field { ϕ i } 1 ≤ i ≤ N under P p 0 , N is distributed like { Z i } 1 ≤ i ≤ N conditionally on ( Y N , Z N ) = (0 , 0). ( Z N / 2 ≈ N 3 / 2 ) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34
The Models Free Energy Path Results Proof Integrated random walk A random walk viewpoint ( ε = 0) Let { X i } i ∈ N be IID random variables with law P ( X i ∈ d x ) := e − V ( x ) d x E ( X i ) = 0 Var ( X i ) = 1 Random walk: Y n := X 1 + . . . + X n Integrated random walk: entropic repulsion Z n := Y 1 + . . . + Y n = n X 1 + ( n − 1) X 2 + . . . + X n The free case ε = 0 ◮ The field { ϕ i } 1 ≤ i ≤ N under P p 0 , N is distributed like { Z i } 1 ≤ i ≤ N conditionally on ( Y N , Z N ) = (0 , 0). ( Z N / 2 ≈ N 3 / 2 ) ◮ Under P w 0 , N the same, under the further conditioning { Z 1 ≥ 0 , . . . , Z N ≥ 0 } Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 28 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Auxiliary chain: J i := ϕ τ i − 1 for i ∈ N Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Auxiliary chain: J i := ϕ τ i − 1 for i ∈ N Conditionally on τ and J , the excursions { e i ( · ) } i ∈ N under P a ε, N are independent and distributed: Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Auxiliary chain: J i := ϕ τ i − 1 for i ∈ N Conditionally on τ and J , the excursions { e i ( · ) } i ∈ N under P a ε, N are independent and distributed: ◮ for a = p like bridges of the process { Z i } i Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Auxiliary chain: J i := ϕ τ i − 1 for i ∈ N Conditionally on τ and J , the excursions { e i ( · ) } i ∈ N under P a ε, N are independent and distributed: ◮ for a = p like bridges of the process { Z i } i ◮ for a = w like bridges of the process { Z i } i conditioned to stay non-negative Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk Excursions and contact set What happens for ε > 0? The same holds for the excursions. � � Contact set: τ := i ∈ N : ϕ i = 0 = { τ i } i ≥ 0 Excursions: { e i ( k ) } k := { ϕ τ i − 1 + k } 0 ≤ k ≤ τ i − τ i − 1 for i ∈ N Auxiliary chain: J i := ϕ τ i − 1 for i ∈ N Conditionally on τ and J , the excursions { e i ( · ) } i ∈ N under P a ε, N are independent and distributed: ◮ for a = p like bridges of the process { Z i } i ◮ for a = w like bridges of the process { Z i } i conditioned to stay non-negative Once we know τ, J , the whole field { ϕ i } i is reconstructed by pasting independent excursions from { Z i } i (cond. to stay ≥ 0) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 29 / 34
The Models Free Energy Path Results Proof Integrated random walk The law of the excursions Pinning case: good control (Donsker’s inv. pr. + LLT) � Z � Nt � � �� � condit. on ( Y N , Z N ) = (0 , 0) = ⇒ I t N 3 / 2 t ∈ [0 , 1] t ∈ [0 , 1] Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34
The Models Free Energy Path Results Proof Integrated random walk The law of the excursions Pinning case: good control (Donsker’s inv. pr. + LLT) � Z � Nt � � �� � condit. on ( Y N , Z N ) = (0 , 0) = ⇒ I t N 3 / 2 t ∈ [0 , 1] t ∈ [0 , 1] Wetting case: several open issues ◮ Integrated BM conditioned to stay non-negative is studied, but not its bridge Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34
The Models Free Energy Path Results Proof Integrated random walk The law of the excursions Pinning case: good control (Donsker’s inv. pr. + LLT) � Z � Nt � � �� � condit. on ( Y N , Z N ) = (0 , 0) = ⇒ I t N 3 / 2 t ∈ [0 , 1] t ∈ [0 , 1] Wetting case: several open issues ◮ Integrated BM conditioned to stay non-negative is studied, but not its bridge ◮ Invariance principle seems in any case very difficult Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34
The Models Free Energy Path Results Proof Integrated random walk The law of the excursions Pinning case: good control (Donsker’s inv. pr. + LLT) � Z � Nt � � �� � condit. on ( Y N , Z N ) = (0 , 0) = ⇒ I t N 3 / 2 t ∈ [0 , 1] t ∈ [0 , 1] Wetting case: several open issues ◮ Integrated BM conditioned to stay non-negative is studied, but not its bridge ◮ Invariance principle seems in any case very difficult Entropic repulsion � � Z 1 ≥ 0 , . . . , Z N ≥ 0 ≈ ? P � � � � Y N = 0 , Z N = 0 Z 1 ≥ 0 , . . . , Z N ≥ 0 ≈ ? P Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34
The Models Free Energy Path Results Proof Integrated random walk The law of the excursions Pinning case: good control (Donsker’s inv. pr. + LLT) � Z � Nt � � �� � condit. on ( Y N , Z N ) = (0 , 0) = ⇒ I t N 3 / 2 t ∈ [0 , 1] t ∈ [0 , 1] Wetting case: several open issues ◮ Integrated BM conditioned to stay non-negative is studied, but not its bridge ◮ Invariance principle seems in any case very difficult Entropic repulsion � � ≈ N − 1 / 4 Z 1 ≥ 0 , . . . , Z N ≥ 0 [Sinai (SRW)] P � � � � Y N = 0 , Z N = 0 ≈ N − 1 / 2 Z 1 ≥ 0 , . . . , Z N ≥ 0 [conj.] P Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 30 / 34
The Models Free Energy Path Results Proof Markov renewal theory Markov renewal processes Given a (sub-)probability kernel K x , d y ( n ): � � K x , d y ( n ) = c ≤ 1 , ∀ x ∈ R y ∈ R n ∈ N we build the Markov renewal process τ with modulating chain J : � � � � P τ i +1 − τ i = n , J i +1 ∈ d y � J i = x := K x , d y ( n ) Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 31 / 34
The Models Free Energy Path Results Proof Markov renewal theory Markov renewal processes Given a (sub-)probability kernel K x , d y ( n ): � � K x , d y ( n ) = c ≤ 1 , ∀ x ∈ R y ∈ R n ∈ N we build the Markov renewal process τ with modulating chain J : � � � � P τ i +1 − τ i = n , J i +1 ∈ d y � J i = x := K x , d y ( n ) The law of ( τ, J ) conditionally on { N , N + 1 } ⊆ τ is � � � � 1 � { N , N + 1 } ⊆ τ P τ i = t i , J i ∈ d y i = K y i − 1 , d y i ( t i − t i − 1 ) C N i with C N = P ( { N , N + 1 } ⊆ τ ). Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 31 / 34
The Models Free Energy Path Results Proof Markov renewal theory The law of the contact set Consider the following kernels: for n ∈ N and x , y ∈ R � � � � Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G p � � d z z =0 � � � � Z i ≥ 0 , i ≤ n , Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G w � � d z z =0 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34
The Models Free Energy Path Results Proof Markov renewal theory The law of the contact set Consider the following kernels: for n ∈ N and x , y ∈ R � � � � Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G p � � d z z =0 � � � � Z i ≥ 0 , i ≤ n , Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G w � � d z z =0 The law of ( τ, J ) is given by k � � � 1 P a G a τ i = t i , J i ∈ d y i , i ≤ k = y i − 1 , d y i ( t i − t i − 1 ) ε, N Z a ε, N i =1 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34
The Models Free Energy Path Results Proof Markov renewal theory The law of the contact set Consider the following kernels: for n ∈ N and x , y ∈ R � � � � Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G p � � d z z =0 � � � � Z i ≥ 0 , i ≤ n , Z n − 1 ∈ d y , Z n ∈ d z x , d y ( n ) := ε P x G w � � d z z =0 The law of ( τ, J ) is given by k � � � 1 P a G a τ i = t i , J i ∈ d y i , i ≤ k = y i − 1 , d y i ( t i − t i − 1 ) ε, N Z a ε, N i =1 Reminds of Markov renewal theory Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 32 / 34
The Models Free Energy Path Results Proof Markov renewal theory Markov renewal processes We exploit the invariance properties: for every F , v ( y ) � � P a τ i = t i , J i ∈ d y i , i ≤ k ε, N � k e F N v ( y i ) y i − 1 , d y i ( t i − t i − 1 ) e − F ( t i − t i − 1 ) G a = Z a v ( y i − 1 ) ε, N i =1 Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction June 11, 2008 33 / 34
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