The Models Free Energy Path Results Proofs
Pinning and Wetting Transition for(1+1)-Dimensional Fieldswith Laplacian Interaction
Francesco Caravenna
Universita degli Studi di Padova
Seminar on Stochastic Processes
Zurich, April 30th, 2008
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 1 / 34
The Models Free Energy Path Results Proofs
References
I [CD1] F. Caravenna and J.-D. DeuschelPinning and wetting transition for (1+1)-dimensional fieldswith Laplacian interaction, Ann. Probab. (to appear)
I [CD2] F. Caravenna and J.-D. DeuschelScaling limits of (1+1)-dimensional pinning models withLaplacian interaction, preprint (2008).
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 2 / 34
The Models Free Energy Path Results Proofs
Outline
1. The ModelsIntroductionWetting and pinning models
2. Free Energy ResultsThe free energyThe phase transitionThe disordered case
3. Path ResultsPath resultsRefined critical scaling limit
4. Sketch of the ProofsIntegrated random walkMarkov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 3 / 34
The Models Free Energy Path Results Proofs
Outline
1. The ModelsIntroductionWetting and pinning models
2. Free Energy ResultsThe free energyThe phase transitionThe disordered case
3. Path ResultsPath resultsRefined critical scaling limit
4. Sketch of the ProofsIntegrated random walkMarkov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 4 / 34
The Models Free Energy Path Results Proofs Introduction
Some motivations
DNA denaturation transition at high temperature
(1+1)-dimensional model: field above an impenetrable wall
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 5 / 34
The Models Free Energy Path Results Proofs Introduction
Some motivations
DNA denaturation transition at high temperature
(1+1)-dimensional model: field above an impenetrable wall
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 5 / 34
The Models Free Energy Path Results Proofs Introduction
Some motivations
DNA denaturation transition at high temperature
(1+1)-dimensional model: field above an impenetrable wall
Energy
Entropy
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 5 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The general wetting model
The field ϕ = ϕi1≤i≤N in the free case:
Pw0,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zw0,N
N∏i=1
(
dϕ+i
+ ε · δ0(dϕi ))
I dϕ+i is the Lebesgue measure on [0,∞)
I HN(ϕ) describes the structure of the chain (to be specified)
I Zw0,N is the normalization constant (partition function)
I δ0(·) is the Dirac mass at zero
I ε ≥ 0 is the strength of the pinning interaction
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 6 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The general wetting model
The field ϕ = ϕi1≤i≤N in the interacting case:
Pwε,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zwε,N
N∏i=1
(dϕ+
i + ε · δ0(dϕi ))
I dϕ+i is the Lebesgue measure on [0,∞)
I HN(ϕ) describes the structure of the chain (to be specified)
I Zwε,N is the normalization constant (partition function)
I δ0(·) is the Dirac mass at zero
I ε ≥ 0 is the strength of the pinning interaction
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 6 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The general pinning model
Analogous to the wetting case but without repulsion: dϕ+i → dϕi
Ppε,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zpε,N
N∏i=1
(dϕi + ε · δ0(dϕi )
)
I dϕi is the Lebesgue measure on RI HN(ϕ) is the same Hamiltonian (to be specified)
I Zpε,N is the normalization constant (partition function)
I δ0(·) is the Dirac mass at zero
I ε ≥ 0 is the strength of the pinning interaction
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 7 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The general pinning model
Analogous to the wetting case but without repulsion: dϕ+i → dϕi
Ppε,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zpε,N
N∏i=1
(dϕi + ε · δ0(dϕi )
)
I dϕi is the Lebesgue measure on RI HN(ϕ) is the same Hamiltonian (to be specified)
I Zpε,N is the normalization constant (partition function)
I δ0(·) is the Dirac mass at zero
I ε ≥ 0 is the strength of the pinning interaction
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 7 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Pinning VS Wetting (+ boundary conditions)(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3
N
N
N + 1
N + 1
!1
!1
0
0
!nn
!nn
pinning model Pp!,N
wetting model Pw!,N
Figure 1. A graphical representation of the pinning model Pp!,N (top)
and of the wetting model Pw!,N (bottom), for N = 25 and " > 0. The
trajectories (n,!n)0!n!N of the field describe the configurations of alinear chain attracted to a defect line, the x–axis. The grey circles representthe pinned sites, i.e. the points in which the chain touches the defect line,which are energetically favored. Note that in the pinning case the chain cancross the defect line without touching it, while this does not happen in thewetting case due to the presence of a wall, i.e. of a constraint for the chainto stay non-negative: the repulsion e!ect of entropic nature that arises isresponsible for the di!erent critical behavior of the models.
1.2. The free energy and the main results. A convenient way to define localizationfor our models is by looking at the Laplace asymptotic behavior of the partition functionZa
!,N as N "#. More precisely, for a $ p,w we define the free energy fa(") by
fa(") := limN"#
faN (") , fa
N (") :=1N
logZa!,N , (1.8)
where the existence of this limit (that will follow as a by-product of our approach) can beproven with a standard super-additivity argument. The basic observation is that the freeenergy is non-negative. In fact, setting "p := [0,#) and "w := R, we have %N $ N
Za!,N =
!exp
"!H[$1,N+1](!)# N$1$
i=1
"" #0(d!i) + d!i 1("i%!a)
#&
!exp
"!H[$1,N+1](!)# N$1$
i=1
d!i 1("i%!a) = Za0,N & c1
N c2,
(1.9)
(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3
N
N
N + 1
N + 1
!1
!1
0
0
!nn
!nn
pinning model Pp!,N
wetting model Pw!,N
Figure 1. A graphical representation of the pinning model Pp!,N (top)
and of the wetting model Pw!,N (bottom), for N = 25 and " > 0. The
trajectories (n,!n)0!n!N of the field describe the configurations of alinear chain attracted to a defect line, the x–axis. The grey circles representthe pinned sites, i.e. the points in which the chain touches the defect line,which are energetically favored. Note that in the pinning case the chain cancross the defect line without touching it, while this does not happen in thewetting case due to the presence of a wall, i.e. of a constraint for the chainto stay non-negative: the repulsion e!ect of entropic nature that arises isresponsible for the di!erent critical behavior of the models.
1.2. The free energy and the main results. A convenient way to define localizationfor our models is by looking at the Laplace asymptotic behavior of the partition functionZa
!,N as N "#. More precisely, for a $ p,w we define the free energy fa(") by
fa(") := limN"#
faN (") , fa
N (") :=1N
logZa!,N , (1.8)
where the existence of this limit (that will follow as a by-product of our approach) can beproven with a standard super-additivity argument. The basic observation is that the freeenergy is non-negative. In fact, setting "p := [0,#) and "w := R, we have %N $ N
Za!,N =
!exp
"!H[$1,N+1](!)# N$1$
i=1
"" #0(d!i) + d!i 1("i%!a)
#&
!exp
"!H[$1,N+1](!)# N$1$
i=1
d!i 1("i%!a) = Za0,N & c1
N c2,
(1.9)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 8 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Pinning VS Wetting (+ boundary conditions)(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3
N
N
N + 1
N + 1
!1
!1
0
0
!nn
!nn
pinning model Pp!,N
wetting model Pw!,N
Figure 1. A graphical representation of the pinning model Pp!,N (top)
and of the wetting model Pw!,N (bottom), for N = 25 and " > 0. The
trajectories (n,!n)0!n!N of the field describe the configurations of alinear chain attracted to a defect line, the x–axis. The grey circles representthe pinned sites, i.e. the points in which the chain touches the defect line,which are energetically favored. Note that in the pinning case the chain cancross the defect line without touching it, while this does not happen in thewetting case due to the presence of a wall, i.e. of a constraint for the chainto stay non-negative: the repulsion e!ect of entropic nature that arises isresponsible for the di!erent critical behavior of the models.
1.2. The free energy and the main results. A convenient way to define localizationfor our models is by looking at the Laplace asymptotic behavior of the partition functionZa
!,N as N "#. More precisely, for a $ p,w we define the free energy fa(") by
fa(") := limN"#
faN (") , fa
N (") :=1N
logZa!,N , (1.8)
where the existence of this limit (that will follow as a by-product of our approach) can beproven with a standard super-additivity argument. The basic observation is that the freeenergy is non-negative. In fact, setting "p := [0,#) and "w := R, we have %N $ N
Za!,N =
!exp
"!H[$1,N+1](!)# N$1$
i=1
"" #0(d!i) + d!i 1("i%!a)
#&
!exp
"!H[$1,N+1](!)# N$1$
i=1
d!i 1("i%!a) = Za0,N & c1
N c2,
(1.9)
(1+1)–DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION 3
N
N
N + 1
N + 1
!1
!1
0
0
!nn
!nn
pinning model Pp!,N
wetting model Pw!,N
Figure 1. A graphical representation of the pinning model Pp!,N (top)
and of the wetting model Pw!,N (bottom), for N = 25 and " > 0. The
trajectories (n,!n)0!n!N of the field describe the configurations of alinear chain attracted to a defect line, the x–axis. The grey circles representthe pinned sites, i.e. the points in which the chain touches the defect line,which are energetically favored. Note that in the pinning case the chain cancross the defect line without touching it, while this does not happen in thewetting case due to the presence of a wall, i.e. of a constraint for the chainto stay non-negative: the repulsion e!ect of entropic nature that arises isresponsible for the di!erent critical behavior of the models.
1.2. The free energy and the main results. A convenient way to define localizationfor our models is by looking at the Laplace asymptotic behavior of the partition functionZa
!,N as N "#. More precisely, for a $ p,w we define the free energy fa(") by
fa(") := limN"#
faN (") , fa
N (") :=1N
logZa!,N , (1.8)
where the existence of this limit (that will follow as a by-product of our approach) can beproven with a standard super-additivity argument. The basic observation is that the freeenergy is non-negative. In fact, setting "p := [0,#) and "w := R, we have %N $ N
Za!,N =
!exp
"!H[$1,N+1](!)# N$1$
i=1
"" #0(d!i) + d!i 1("i%!a)
#&
!exp
"!H[$1,N+1](!)# N$1$
i=1
d!i 1("i%!a) = Za0,N & c1
N c2,
(1.9)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 8 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Some questions
Once HN(ϕ) is chosen and ε ≥ 0 is fixed:
I What are the properties of Ppε,N and Pw
ε,N for large N ?
I Is the field localized at ϕ = 0 or delocalized in ϕ 6= 0?(How to define localization and delocalization?)
I Does the answer depend on ε ≥ 0 (phase transition) and/oron the choice of HN(ϕ)?
How to choose HN(ϕ) ?
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 9 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Some questions
Once HN(ϕ) is chosen and ε ≥ 0 is fixed:
I What are the properties of Ppε,N and Pw
ε,N for large N ?
I Is the field localized at ϕ = 0 or delocalized in ϕ 6= 0?(How to define localization and delocalization?)
I Does the answer depend on ε ≥ 0 (phase transition) and/oron the choice of HN(ϕ)?
How to choose HN(ϕ) ?
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 9 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Some questions
Once HN(ϕ) is chosen and ε ≥ 0 is fixed:
I What are the properties of Ppε,N and Pw
ε,N for large N ?
I Is the field localized at ϕ = 0 or delocalized in ϕ 6= 0?(How to define localization and delocalization?)
I Does the answer depend on ε ≥ 0 (phase transition) and/oron the choice of HN(ϕ)?
How to choose HN(ϕ) ?
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 9 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Some questions
Once HN(ϕ) is chosen and ε ≥ 0 is fixed:
I What are the properties of Ppε,N and Pw
ε,N for large N ?
I Is the field localized at ϕ = 0 or delocalized in ϕ 6= 0?(How to define localization and delocalization?)
I Does the answer depend on ε ≥ 0 (phase transition) and/oron the choice of HN(ϕ)?
How to choose HN(ϕ) ?
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 9 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
The simplest choice is the gradient case:
HN(ϕ) :=N∑
i=1
V(∇ϕi
), ∇ϕi := ϕi − ϕi−1 ,
V (·) : R→ R ∪ +∞ :
∫R
e−V (x) dx < +∞ ( = 1 ) .
+ regularity (see later). Gaussian case: V (x) ∝ x2
I Pp0,N is (the bridge of) a random walk of step law e−V (x)
I Pw0,N is (the bridge of) a random walk conditioned to stay ≥ 0
[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05]
[Caravenna, Giacomin, Zambotti EJP 06]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 10 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
The simplest choice is the gradient case:
HN(ϕ) :=N∑
i=1
V(∇ϕi
), ∇ϕi := ϕi − ϕi−1 ,
V (·) : R→ R ∪ +∞ :
∫R
e−V (x) dx < +∞ ( = 1 ) .
+ regularity (see later). Gaussian case: V (x) ∝ x2
I Pp0,N is (the bridge of) a random walk of step law e−V (x)
I Pw0,N is (the bridge of) a random walk conditioned to stay ≥ 0
[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05]
[Caravenna, Giacomin, Zambotti EJP 06]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 10 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
The simplest choice is the gradient case:
HN(ϕ) :=N∑
i=1
V(∇ϕi
), ∇ϕi := ϕi − ϕi−1 ,
V (·) : R→ R ∪ +∞ :
∫R
e−V (x) dx < +∞ ( = 1 ) .
+ regularity (see later). Gaussian case: V (x) ∝ x2
I Pp0,N is (the bridge of) a random walk of step law e−V (x)
I Pw0,N is (the bridge of) a random walk conditioned to stay ≥ 0
[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05]
[Caravenna, Giacomin, Zambotti EJP 06]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 10 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
The simplest choice is the gradient case:
HN(ϕ) :=N∑
i=1
V(∇ϕi
), ∇ϕi := ϕi − ϕi−1 ,
V (·) : R→ R ∪ +∞ :
∫R
e−V (x) dx < +∞ ( = 1 ) .
+ regularity (see later). Gaussian case: V (x) ∝ x2
I Pp0,N is (the bridge of) a random walk of step law e−V (x)
I Pw0,N is (the bridge of) a random walk conditioned to stay ≥ 0
[Isozaki, Yoshida SPA 01] [Deuschel, Giacomin, Zambotti PTRF 05]
[Caravenna, Giacomin, Zambotti EJP 06]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 10 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
We rather consider the Laplacian case:
HN(ϕ) :=N∑
i=0
V(∆ϕi
)
+ V(∇ϕi
)
∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi
I Simplest non-nearest neighbor interaction
I Semi-flexible polymers: ∆ favors affine configurations
I ∇ and ∆ together?
Interpretation of the free case ε = 0:
I Pp0,N is (the bridge of) the integral of a random walk
I Pw0,N is further conditioned to stay ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 11 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
We rather consider the Laplacian case:
HN(ϕ) :=N∑
i=0
V(∆ϕi
)
+ V(∇ϕi
)
∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi
I Simplest non-nearest neighbor interaction
I Semi-flexible polymers: ∆ favors affine configurations
I ∇ and ∆ together?
Interpretation of the free case ε = 0:
I Pp0,N is (the bridge of) the integral of a random walk
I Pw0,N is further conditioned to stay ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 11 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
We rather consider the Laplacian case:
HN(ϕ) :=N∑
i=0
V(∆ϕi
)+ V
(∇ϕi
)∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi
I Simplest non-nearest neighbor interaction
I Semi-flexible polymers: ∆ favors affine configurations
I ∇ and ∆ together?
Interpretation of the free case ε = 0:
I Pp0,N is (the bridge of) the integral of a random walk
I Pw0,N is further conditioned to stay ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 11 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
The choice of HN(ϕ)
We rather consider the Laplacian case:
HN(ϕ) :=N∑
i=0
V(∆ϕi
)
+ V(∇ϕi
)
∆ϕi := ∇ϕi+1 −∇ϕi = ϕi+1 + ϕi−1 − 2ϕi
I Simplest non-nearest neighbor interaction
I Semi-flexible polymers: ∆ favors affine configurations
I ∇ and ∆ together?
Interpretation of the free case ε = 0:
I Pp0,N is (the bridge of) the integral of a random walk
I Pw0,N is further conditioned to stay ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 11 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Laplacian interaction in (d + 1)-dimension
Fields ϕ : 1, . . . ,Nd → R with Laplacian interaction for d ≥ 2are models for semiflexible membranes
The problem of entropic repulsion (probability for the field to staynon-negative) has been studied in the Gaussian case:
[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]
Henceforth we study Ppε,N and Pw
ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0Assumptions on V :∫
Re−V (x) dx = 1 ,
∫R
x e−V (x) dx = 0 ,
∫R
x2 e−V (x) dx = 1
+ regularity: x 7→ e−V (x) continuous and V (0) < +∞.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 12 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Laplacian interaction in (d + 1)-dimension
Fields ϕ : 1, . . . ,Nd → R with Laplacian interaction for d ≥ 2are models for semiflexible membranes
The problem of entropic repulsion (probability for the field to staynon-negative) has been studied in the Gaussian case:
[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]
Henceforth we study Ppε,N and Pw
ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0Assumptions on V :∫
Re−V (x) dx = 1 ,
∫R
x e−V (x) dx = 0 ,
∫R
x2 e−V (x) dx = 1
+ regularity: x 7→ e−V (x) continuous and V (0) < +∞.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 12 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Laplacian interaction in (d + 1)-dimension
Fields ϕ : 1, . . . ,Nd → R with Laplacian interaction for d ≥ 2are models for semiflexible membranes
The problem of entropic repulsion (probability for the field to staynon-negative) has been studied in the Gaussian case:
[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]
Henceforth we study Ppε,N and Pw
ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0
Assumptions on V :∫R
e−V (x) dx = 1 ,
∫R
x e−V (x) dx = 0 ,
∫R
x2 e−V (x) dx = 1
+ regularity: x 7→ e−V (x) continuous and V (0) < +∞.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 12 / 34
The Models Free Energy Path Results Proofs Wetting and pinning models
Laplacian interaction in (d + 1)-dimension
Fields ϕ : 1, . . . ,Nd → R with Laplacian interaction for d ≥ 2are models for semiflexible membranes
The problem of entropic repulsion (probability for the field to staynon-negative) has been studied in the Gaussian case:
[Sakagawa JMP 04] [Kurt SPA 07] [Kurt preprint 07]
Henceforth we study Ppε,N and Pw
ε,N with Laplacian interaction andboundary conditions ϕ−1 = ϕ0 = ϕN = ϕN+1 = 0Assumptions on V :∫
Re−V (x) dx = 1 ,
∫R
x e−V (x) dx = 0 ,
∫R
x2 e−V (x) dx = 1
+ regularity: x 7→ e−V (x) continuous and V (0) < +∞.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 12 / 34
The Models Free Energy Path Results Proofs
Outline
1. The ModelsIntroductionWetting and pinning models
2. Free Energy ResultsThe free energyThe phase transitionThe disordered case
3. Path ResultsPath resultsRefined critical scaling limit
4. Sketch of the ProofsIntegrated random walkMarkov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 13 / 34
The Models Free Energy Path Results Proofs The free energy
How to define localization and delocalization?
Recall the partition function: (zero boundary conditions)
Zaε,N =
∫Ωa
N
e−HN(ϕ)N−1∏i=1
(dϕi + ε δ0(dϕi )
)where a ∈ p,w and Ωp
N = RN−1 while ΩwN = [0,∞)N−1
Free Energy
Fa(ε) := limN→∞
1
Nlog Za
ε,N (super-additivity)
Basic observation: Fa(ε) ≥ Fa(0) = 0 for all ε ≥ 0 and a ∈ p,w
Zaε,N ≥ Za
0,N ≈ N−c (c > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 14 / 34
The Models Free Energy Path Results Proofs The free energy
How to define localization and delocalization?
Recall the partition function: (zero boundary conditions)
Zaε,N =
∫Ωa
N
e−HN(ϕ)N−1∏i=1
(dϕi + ε δ0(dϕi )
)where a ∈ p,w and Ωp
N = RN−1 while ΩwN = [0,∞)N−1
Free Energy
Fa(ε) := limN→∞
1
Nlog Za
ε,N (super-additivity)
Basic observation: Fa(ε) ≥ Fa(0) = 0 for all ε ≥ 0 and a ∈ p,w
Zaε,N ≥ Za
0,N ≈ N−c (c > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 14 / 34
The Models Free Energy Path Results Proofs The free energy
How to define localization and delocalization?
Recall the partition function: (zero boundary conditions)
Zaε,N =
∫Ωa
N
e−HN(ϕ)N−1∏i=1
(dϕi + ε δ0(dϕi )
)where a ∈ p,w and Ωp
N = RN−1 while ΩwN = [0,∞)N−1
Free Energy
Fa(ε) := limN→∞
1
Nlog Za
ε,N (super-additivity)
Basic observation: Fa(ε) ≥ Fa(0) = 0 for all ε ≥ 0 and a ∈ p,w
Zaε,N ≥ Za
0,N ≈ N−c (c > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 14 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The free energy
Localization and delocalization
Definition
The a-model at ε ≥ 0 is said to be localized if Fa(ε) > 0.
localized ⇐⇒ ε > εac := supε ≥ 0 : Fa(ε) = 0 ∈ [0,∞]
Setting `N := #
i ≤ N : ϕi = 0
we have:
I if ε > εac then`NN→ Da(ε) > 0 in Pa
ε,N –probability
I if ε < εac then`NN→ 0 in Pa
ε,N –probability (delocalization)
I if ε = εac ? Depends on the model:
I if Fa(εac + h) = o(h) [> 1st order trans.] ε = εac is delocalized
I if Fa(εac + h) ≥ C h [1st order trans.] ε = εac may be localized(phase coexistence, dependence of boundary conditions)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 15 / 34
The Models Free Energy Path Results Proofs The phase transition
The phase transition
Theorem ([CD1])
Both Ppε,N and Pw
ε,N undergo a non-trivial phase transition:
0 < εpc < εwc < ∞
and Fa(ε) is analytic on [0, εac) ∪ (εac ,∞). (variational formula)
I In the pinning model the transition is exactly of 2nd order:
C1h
log 1h
≤ Fp(εpc + h) ≤ o(h)
I In the wetting model the transition is of 1st order:
Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 16 / 34
The Models Free Energy Path Results Proofs The phase transition
The phase transition
Theorem ([CD1])
Both Ppε,N and Pw
ε,N undergo a non-trivial phase transition:
0 < εpc < εwc < ∞
and Fa(ε) is analytic on [0, εac) ∪ (εac ,∞). (variational formula)
I In the pinning model the transition is exactly of 2nd order:
C1h
log 1h
≤ Fp(εpc + h) ≤ o(h)
I In the wetting model the transition is of 1st order:
Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 16 / 34
The Models Free Energy Path Results Proofs The phase transition
The phase transition
Theorem ([CD1])
Both Ppε,N and Pw
ε,N undergo a non-trivial phase transition:
0 < εpc < εwc < ∞
and Fa(ε) is analytic on [0, εac) ∪ (εac ,∞). (variational formula)
I In the pinning model the transition is exactly of 2nd order:
C1h
log 1h
≤ Fp(εpc + h) ≤ o(h)
I In the wetting model the transition is of 1st order:
Fw(εwc + h) ∼ C2 h , `N ∼ DN (D > 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 16 / 34
The Models Free Energy Path Results Proofs The phase transition
The gradient case
Differences in the gradient case
I the transition is non-trivial only in the wetting model:
εp,∇c = 0 , 0 < εw,∇c < ∞
I the transition is of 2nd order:
Fp(εp,∇c + h
) ∼ Cp h2 , Fw(εw,∇c + h
) ∼ Cw h2
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
%
"
""pc"p,"
c
Fp(")
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
4 FRANCESCO CARAVENNA
!
!
!!wc!w,!
c
Fw(!)
Figure 2. Write caption.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 17 / 34
The Models Free Energy Path Results Proofs The phase transition
The gradient case
Differences in the gradient case
I the transition is non-trivial only in the wetting model:
εp,∇c = 0 , 0 < εw,∇c < ∞I the transition is of 2nd order:
Fp(εp,∇c + h
) ∼ Cp h2 , Fw(εw,∇c + h
) ∼ Cw h2
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
%
"
""pc"p,"
c
Fp(")
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
4 FRANCESCO CARAVENNA
!
!
!!wc!w,!
c
Fw(!)
Figure 2. Write caption.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 17 / 34
The Models Free Energy Path Results Proofs The phase transition
The gradient case
Differences in the gradient case
I the transition is non-trivial only in the wetting model:
εp,∇c = 0 , 0 < εw,∇c < ∞I the transition is of 2nd order:
Fp(εp,∇c + h
) ∼ Cp h2 , Fw(εw,∇c + h
) ∼ Cw h2
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
%
"
""pc"p,"
c
Fp(")
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
4 FRANCESCO CARAVENNA
!
!
!!wc!w,!
c
Fw(!)
Figure 2. Write caption.Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 17 / 34
The Models Free Energy Path Results Proofs The disordered case
A look at the disordered case
Disordered version of our model: (dϕpi = dϕi and dϕw
i = dϕ+i )
Paε,β,ω,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zaε,β,ω,N
N∏i=1
(dϕa
i + ε eβωi δ0(dϕi ))
where β ≥ 0 and ωii∈N are IID N (0, 1) (law P indep. Pa).
Quenched free energy
Fa(ε, β)
:= limN→∞
1
Nlog Za
ε,β,ω,N ≥ 0
exists P(dω)–a.s. and does not depend on ω (self-averaging)
Localization: Fa(ε, β) > 0 ⇐⇒ ε > εac(β) (critical line)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 18 / 34
The Models Free Energy Path Results Proofs The disordered case
A look at the disordered case
Disordered version of our model: (dϕpi = dϕi and dϕw
i = dϕ+i )
Paε,β,ω,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zaε,β,ω,N
N∏i=1
(dϕa
i + ε eβωi δ0(dϕi ))
where β ≥ 0 and ωii∈N are IID N (0, 1) (law P indep. Pa).
Quenched free energy
Fa(ε, β)
:= limN→∞
1
Nlog Za
ε,β,ω,N ≥ 0
exists P(dω)–a.s. and does not depend on ω (self-averaging)
Localization: Fa(ε, β) > 0 ⇐⇒ ε > εac(β) (critical line)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 18 / 34
The Models Free Energy Path Results Proofs The disordered case
A look at the disordered case
Disordered version of our model: (dϕpi = dϕi and dϕw
i = dϕ+i )
Paε,β,ω,N
(dϕ1 , . . . , dϕN
):=
e−HN(ϕ)
Zaε,β,ω,N
N∏i=1
(dϕa
i + ε eβωi δ0(dϕi ))
where β ≥ 0 and ωii∈N are IID N (0, 1) (law P indep. Pa).
Quenched free energy
Fa(ε, β)
:= limN→∞
1
Nlog Za
ε,β,ω,N ≥ 0
exists P(dω)–a.s. and does not depend on ω (self-averaging)
Localization: Fa(ε, β) > 0 ⇐⇒ ε > εac(β) (critical line)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 18 / 34
The Models Free Energy Path Results Proofs The disordered case
Smoothing effect of disorder
What is the behavior of εac(β) for small β ? (εac = εac(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 Cβ > 0 such that
Fa(εac(β) + h
) ≤ Cβ h2
When disorder is present the transition is at least of 2nd orderSharp contrast with homogeneous case cf.
Very general proof: rare-stretches in ω (Large Deviations)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 19 / 34
The Models Free Energy Path Results Proofs The disordered case
Smoothing effect of disorder
What is the behavior of εac(β) for small β ? (εac = εac(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 Cβ > 0 such that
Fa(εac(β) + h
) ≤ Cβ h2
When disorder is present the transition is at least of 2nd orderSharp contrast with homogeneous case cf.
Very general proof: rare-stretches in ω (Large Deviations)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 19 / 34
The Models Free Energy Path Results Proofs The disordered case
Smoothing effect of disorder
What is the behavior of εac(β) for small β ? (εac = εac(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 Cβ > 0 such that
Fa(εac(β) + h
) ≤ Cβ h2
When disorder is present the transition is at least of 2nd orderSharp contrast with homogeneous case cf.
Very general proof: rare-stretches in ω (Large Deviations)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 19 / 34
The Models Free Energy Path Results Proofs
Outline
1. The ModelsIntroductionWetting and pinning models
2. Free Energy ResultsThe free energyThe phase transitionThe disordered case
3. Path ResultsPath resultsRefined critical scaling limit
4. Sketch of the ProofsIntegrated random walkMarkov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 20 / 34
The Models Free Energy Path Results Proofs
Some deeper questions
We have established the existence of a phase transition:
`N = o(N) if ε < εac VS `N ∼ D · N if ε > εac
Can we say something more precise? Typical paths of Paε,N ?
Yes in the pinning case and under additional assumptions on V (·):
I symmetry: V (x) = V (−x) for every x ∈ RI uniform strict convexity: ∃γ > 0 s. t. V (x)− γ x2
2 is convex
I regularity: x 7→ e−V (x) is continuous and V (0) <∞∫R
e−V (x) dx = 1
∫R
x2 e−V (x) dx = 1
Let us set MN := max1≤i≤N
∣∣ϕi
∣∣
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 21 / 34
The Models Free Energy Path Results Proofs
Some deeper questions
We have established the existence of a phase transition:
`N = o(N) if ε < εac VS `N ∼ D · N if ε > εac
Can we say something more precise? Typical paths of Paε,N ?
Yes in the pinning case and under additional assumptions on V (·):
I symmetry: V (x) = V (−x) for every x ∈ RI uniform strict convexity: ∃γ > 0 s. t. V (x)− γ x2
2 is convex
I regularity: x 7→ e−V (x) is continuous and V (0) <∞∫R
e−V (x) dx = 1
∫R
x2 e−V (x) dx = 1
Let us set MN := max1≤i≤N
∣∣ϕi
∣∣
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 21 / 34
The Models Free Energy Path Results Proofs
Some deeper questions
We have established the existence of a phase transition:
`N = o(N) if ε < εac VS `N ∼ D · N if ε > εac
Can we say something more precise? Typical paths of Paε,N ?
Yes in the pinning case and under additional assumptions on V (·):
I symmetry: V (x) = V (−x) for every x ∈ RI uniform strict convexity: ∃γ > 0 s. t. V (x)− γ x2
2 is convex
I regularity: x 7→ e−V (x) is continuous and V (0) <∞∫R
e−V (x) dx = 1
∫R
x2 e−V (x) dx = 1
Let us set MN := max1≤i≤N
∣∣ϕi
∣∣Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 21 / 34
The Models Free Energy Path Results Proofs Path results
A closer look at the typical paths [CD2]
I delocalized regime ε < εpc : `N = O(1) and MN ≈ N3/2 :
limK→∞
lim infN→∞
Ppε,N
(ϕi 6= 0 for i ∈ K ,N − K) = 1
limK→∞
lim infN→∞
Ppε,N
(1
KN3/2 ≤ MN ≤ K N3/2
)= 1
I localized regime ε > εpc : MN = O((log N)2
):
limK→∞
lim infN→∞
Ppε,N
(MN ≤ K (log N)2
)= 1
I critical regime ε = εpc : MN ≈ N3/2
(log N)c and `N ≈ Nlog N :
limK→∞
lim infN→∞
Ppεpc ,N
(1
K
N3/2
(log N)3/2≤ MN ≤ K
N3/2
log N
)= 1
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 22 / 34
The Models Free Energy Path Results Proofs Path results
A closer look at the typical paths [CD2]
I delocalized regime ε < εpc : `N = O(1) and MN ≈ N3/2 :
limK→∞
lim infN→∞
Ppε,N
(ϕi 6= 0 for i ∈ K ,N − K) = 1
limK→∞
lim infN→∞
Ppε,N
(1
KN3/2 ≤ MN ≤ K N3/2
)= 1
I localized regime ε > εpc : MN = O((log N)2
):
limK→∞
lim infN→∞
Ppε,N
(MN ≤ K (log N)2
)= 1
I critical regime ε = εpc : MN ≈ N3/2
(log N)c and `N ≈ Nlog N :
limK→∞
lim infN→∞
Ppεpc ,N
(1
K
N3/2
(log N)3/2≤ MN ≤ K
N3/2
log N
)= 1
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 22 / 34
The Models Free Energy Path Results Proofs Path results
A closer look at the typical paths [CD2]
I delocalized regime ε < εpc : `N = O(1) and MN ≈ N3/2 :
limK→∞
lim infN→∞
Ppε,N
(ϕi 6= 0 for i ∈ K ,N − K) = 1
limK→∞
lim infN→∞
Ppε,N
(1
KN3/2 ≤ MN ≤ K N3/2
)= 1
I localized regime ε > εpc : MN = O((log N)2
):
limK→∞
lim infN→∞
Ppε,N
(MN ≤ K (log N)2
)= 1
I critical regime ε = εpc : MN ≈ N3/2
(log N)c and `N ≈ Nlog N :
limK→∞
lim infN→∞
Ppεpc ,N
(1
K
N3/2
(log N)3/2≤ MN ≤ K
N3/2
log N
)= 1
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 22 / 34
The Models Free Energy Path Results Proofs Path results
Scaling Limits
We rescale and interpolate linearly the field: for t ∈ [0, 1]
ϕN(t) :=ϕbNtcN3/2
+ (Nt − bNtc) ϕbNtc+1 − ϕbNtcN3/2
Let Btt∈[0,1] standard BM, It :=∫ t
0 Bs ds integrated BM(Bt , It)
t∈[0,1]
=
(Bt , It)
t∈[0,1]cond. on (B1, I1) = (0, 0)
Theorem (Scaling Limits [CD2])
The rescaled field ϕN(t)t∈[0,1] under Ppε,N converges in
distribution on C ([0, 1]) as N →∞, for every ε ≥ 0. The limit is
I If ε < εpc , the law of Itt∈[0,1]
I If ε = εpc or ε > εpc , the law concentrated on f (t) ≡ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 23 / 34
The Models Free Energy Path Results Proofs Path results
Scaling Limits
We rescale and interpolate linearly the field: for t ∈ [0, 1]
ϕN(t) :=ϕbNtcN3/2
+ (Nt − bNtc) ϕbNtc+1 − ϕbNtcN3/2
Let Btt∈[0,1] standard BM, It :=∫ t
0 Bs ds integrated BM(Bt , It)
t∈[0,1]
=
(Bt , It)
t∈[0,1]cond. on (B1, I1) = (0, 0)
Theorem (Scaling Limits [CD2])
The rescaled field ϕN(t)t∈[0,1] under Ppε,N converges in
distribution on C ([0, 1]) as N →∞, for every ε ≥ 0. The limit is
I If ε < εpc , the law of Itt∈[0,1]
I If ε = εpc or ε > εpc , the law concentrated on f (t) ≡ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 23 / 34
The Models Free Energy Path Results Proofs Path results
Scaling Limits
We rescale and interpolate linearly the field: for t ∈ [0, 1]
ϕN(t) :=ϕbNtcN3/2
+ (Nt − bNtc) ϕbNtc+1 − ϕbNtcN3/2
Let Btt∈[0,1] standard BM, It :=∫ t
0 Bs ds integrated BM(Bt , It)
t∈[0,1]
=
(Bt , It)
t∈[0,1]cond. on (B1, I1) = (0, 0)
Theorem (Scaling Limits [CD2])
The rescaled field ϕN(t)t∈[0,1] under Ppε,N converges in
distribution on C ([0, 1]) as N →∞, for every ε ≥ 0. The limit is
I If ε < εpc , the law of Itt∈[0,1]
I If ε = εpc or ε > εpc , the law concentrated on f (t) ≡ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 23 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
For ε = εpc the field has very large fluctuations (≈ N3/2
(log N)c ).Can we extract a non-trivial scaling limit?
Not in C ([0, 1]) or D([0, 1]):
1
N
i ∈ 1, . . . ,N : ϕi = 0
becomes dense in [0, 1]
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
!i
O
!N
log N
"
0 N
O
#N3/2
(log N)3/2
$
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 24 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
For ε = εpc the field has very large fluctuations (≈ N3/2
(log N)c ).Can we extract a non-trivial scaling limit?
Not in C ([0, 1]) or D([0, 1]):
1
N
i ∈ 1, . . . ,N : ϕi = 0
becomes dense in [0, 1]
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
!i
O
!N
log N
"
0 N
O
#N3/2
(log N)3/2
$
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 24 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
For ε = εpc the field has very large fluctuations (≈ N3/2
(log N)c ).Can we extract a non-trivial scaling limit?
Not in C ([0, 1]) or D([0, 1]):
1
N
i ∈ 1, . . . ,N : ϕi = 0
becomes dense in [0, 1]
INSERT SHORTER TITLE (IF TITLE IS TOO LONG)! 3
BestJ-DConcerning the repulsion, I wonder whether a simple FKG argument would work:
Consider the gaussian field Ax, x = 0, 1, 2, ... where Ax = S1 + ...Sx and Sx =X1 + ...Xx Xi are i.i.d. gaussian. Note that cov(Ax, Ay) ! 0. That is the field Axsatisfies the FKG property. Let P+
N = P (·|AN!1 ! 0, AN ! 0). Now assume that wecan show that P+
N is also FKG. Let !+N = Sx ! 0, x = 1, 2, ..., N, and for ! > 0,
!!N (!) = SN!1, SN " !. By FKG
P+N (!+
N # !!N(!)) " P+N (!+
N )P+N (!!N (!))
Thus
P (S1 ! 0, S2 ! 0, ...SN2 ! 0, 0 " SN!1 " !, 0 " SN " !)
P (0 " SN!1 " !, 0 " SN " !)" P (!+
N)
P (SN!1 ! 0, SN ! 0)" 4P (!+
N)
since again by FKG
P (SN!1 ! 0, SN ! 0) ! P (SN!1 ! 0)P (SN ! 0) ! 1
4.
Now, letting !$ 0 we are done.In order to prove FKG for P+
N , we can use the fact that a ”product conditioning”does not destroy FKG, but it is a bit subtle, so I have to think about it !
BestJ-D
!i
O
!N
log N
"
0 N
O
#N3/2
(log N)3/2
$
Figure 1. Write caption.
References
[1] W. Feller (1968) An Introduction to Probability Theory and Its Applications, vol. 1, 3rd ed.,John Wiley and Sons, New York
Dipartimento di Matematica Pura e Applicata, Universita di Padova, via Trieste63, 35121 Padova, Italy
E-mail address: [email protected]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 24 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
Alternative idea: look at the field in a distributional sense
µN
(dt)
:=(log N)5/2
N3/2ϕbNtc dt = ϕN(t) dt
µN(·) is a (random) finite signed measure on [0, 1]
Let Ltt∈[0,1] be the stable symmetric Levy process of index 25
0 drift 0 Brownian component Π(dx) =c
|x |7/5dx
The paths of L are a.s. of bounded variation, hence we set
dL((a, b)
):= Lb − La
Theorem ([CD2])
The random signed measure µN under Ppεpc ,N
converges in
distribution as N →∞ toward dL .
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 25 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
Alternative idea: look at the field in a distributional sense
µN
(dt)
:=(log N)5/2
N3/2ϕbNtc dt = ϕN(t) dt
µN(·) is a (random) finite signed measure on [0, 1]
Let Ltt∈[0,1] be the stable symmetric Levy process of index 25
0 drift 0 Brownian component Π(dx) =c
|x |7/5dx
The paths of L are a.s. of bounded variation, hence we set
dL((a, b)
):= Lb − La
Theorem ([CD2])
The random signed measure µN under Ppεpc ,N
converges in
distribution as N →∞ toward dL .
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 25 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
Alternative idea: look at the field in a distributional sense
µN
(dt)
:=(log N)5/2
N3/2ϕbNtc dt = ϕN(t) dt
µN(·) is a (random) finite signed measure on [0, 1]
Let Ltt∈[0,1] be the stable symmetric Levy process of index 25
0 drift 0 Brownian component Π(dx) =c
|x |7/5dx
The paths of L are a.s. of bounded variation, hence we set
dL((a, b)
):= Lb − La
Theorem ([CD2])
The random signed measure µN under Ppεpc ,N
converges in
distribution as N →∞ toward dL .
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 25 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime
Alternative idea: look at the field in a distributional sense
µN
(dt)
:=(log N)5/2
N3/2ϕbNtc dt = ϕN(t) dt
µN(·) is a (random) finite signed measure on [0, 1]
Let Ltt∈[0,1] be the stable symmetric Levy process of index 25
0 drift 0 Brownian component Π(dx) =c
|x |7/5dx
The paths of L are a.s. of bounded variation, hence we set
dL((a, b)
):= Lb − La
Theorem ([CD2])
The random signed measure µN under Ppεpc ,N
converges in
distribution as N →∞ toward dL .
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 25 / 34
The Models Free Energy Path Results Proofs Refined critical scaling limit
The critical regime6 FRANCESCO CARAVENNA AND JEAN–DOMINIQUE DEUSCHEL
0
0 1
1
!!N (t)
dL
O
"1
log N
# O$log N
%
Figure 1. A graphical representation of Theorem 1.6. For large N , theexcursions of the rescaled field under the critical law P!c,N contribute to themeasure µN (dt), see (1.18), approximately like Dirac masses, with intensitygiven by their (signed) area. The width and height of the relevant excursionsare of order (1/ log N) and log N respectively. We warn the reader that thex- and y-axis in the picture have di!erent units of length, and that the fieldcan actually cross the x-axis without touching it (though this feature hasnot been evidenced in the picture for simplicity).
We look at µN under the critical law P!c,N as a random element of M([0, 1]), the space ofall finite signed Borel measures on the interval [0, 1], that we equip with the topology ofvague convergence and with the corresponding Borel "–field (#n ! # vaguely if and onlyif
&fd#n !
&fd# for all bounded and continuous functions f : [0, 1] ! R). Our goal is
to show that the sequence µNN has a non-trivial limit in distribution on M([0, 1]).To describe the limit, let Ltt!0 denote the stable symmetric Levy process of index 2/5
(a standard version with cadlag paths). More explicitly, Ltt!0 is a Levy process with zerodrift, zero Brownian component and with Levy measure given by "(dx) = cL |x|"7/5dx,where the positive constant cL is defined explicitly in equation (6.25). Since the index isless than 1, the paths of L are a.s. of bounded variation, cf. [2], hence we can define pathby path the (random) finite signed measure dL in the Steltjes sense:
dL$(a, b]
%:= Lb " La .
We stress that dL is a.s. a purely atomic measure, i.e., a sum of Dirac masses (for moredetails and for an explicit construction of dL, see Remark 1.7 below).
We are now ready to state our main result (see Figure 1 for a graphical description).
Theorem 1.6. The random signed measure µN under P!c,N converges in distribution onM([0, 1]) as N ! # toward the the random signed measure dL.
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 26 / 34
The Models Free Energy Path Results Proofs
Outline
1. The ModelsIntroductionWetting and pinning models
2. Free Energy ResultsThe free energyThe phase transitionThe disordered case
3. Path ResultsPath resultsRefined critical scaling limit
4. Sketch of the ProofsIntegrated random walkMarkov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 27 / 34
The Models Free Energy Path Results Proofs Integrated random walk
A random walk viewpoint (ε = 0)
Let Xii∈N be IID random variables with law
P(Xi ∈ dx) := e−V (x) dx E(Xi ) = 0 Var(Xi ) = 1
Random walk: Yn := X1 + . . .+ Xn
Integrated random walk:
Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn
The free case ε = 0
I The field ϕi1≤i≤N under Pp0,N is distributed like Zi1≤i≤N
conditionally on (YN ,ZN) = (0, 0). (ZN/2 ≈ N3/2)
I Under Pw0,N the same, under the further conditioning
Z1 ≥ 0, . . . ,ZN ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 28 / 34
The Models Free Energy Path Results Proofs Integrated random walk
A random walk viewpoint (ε = 0)
Let Xii∈N be IID random variables with law
P(Xi ∈ dx) := e−V (x) dx E(Xi ) = 0 Var(Xi ) = 1
Random walk: Yn := X1 + . . .+ Xn
Integrated random walk:
Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn
The free case ε = 0
I The field ϕi1≤i≤N under Pp0,N is distributed like Zi1≤i≤N
conditionally on (YN ,ZN) = (0, 0). (ZN/2 ≈ N3/2)
I Under Pw0,N the same, under the further conditioning
Z1 ≥ 0, . . . ,ZN ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 28 / 34
The Models Free Energy Path Results Proofs Integrated random walk
A random walk viewpoint (ε = 0)
Let Xii∈N be IID random variables with law
P(Xi ∈ dx) := e−V (x) dx E(Xi ) = 0 Var(Xi ) = 1
Random walk: Yn := X1 + . . .+ Xn
Integrated random walk:
Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn
The free case ε = 0
I The field ϕi1≤i≤N under Pp0,N is distributed like Zi1≤i≤N
conditionally on (YN ,ZN) = (0, 0). (ZN/2 ≈ N3/2)
I Under Pw0,N the same, under the further conditioning
Z1 ≥ 0, . . . ,ZN ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 28 / 34
The Models Free Energy Path Results Proofs Integrated random walk
A random walk viewpoint (ε = 0)
Let Xii∈N be IID random variables with law
P(Xi ∈ dx) := e−V (x) dx E(Xi ) = 0 Var(Xi ) = 1
Random walk: Yn := X1 + . . .+ Xn
Integrated random walk:
Zn := Y1 + . . . + Yn = n X1 + (n − 1) X2 + . . . + Xn
The free case ε = 0
I The field ϕi1≤i≤N under Pp0,N is distributed like Zi1≤i≤N
conditionally on (YN ,ZN) = (0, 0). (ZN/2 ≈ N3/2)
I Under Pw0,N the same, under the further conditioning
Z1 ≥ 0, . . . ,ZN ≥ 0
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 28 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process Zii
I for a = w like bridges of the process Zii conditioned to staynon-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
Excursions and contact set
What happens for ε > 0? The same holds for the excursions.
Contact set: τ :=
i ∈ N : ϕi = 0
= τii≥0
Excursions: ei (k)k := ϕτi−1+k0≤k≤τi−τi−1for i ∈ N
Auxiliary chain: Ji := ϕτi−1 for i ∈ N
Conditionally on τ and J, the excursions ei (·)i∈N under Paε,N are
independent and distributed:
I for a = p like bridges of the process ZiiI for a = w like bridges of the process Zii conditioned to stay
non-negative
Once we know τ, J, the whole field ϕii is reconstructed bypasting independent excursions from Zii (cond. to stay ≥ 0)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 29 / 34
The Models Free Energy Path Results Proofs Integrated random walk
The law of the excursions
Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2
t∈[0,1]
condit. on (YN ,ZN) = (0, 0) =⇒
It
t∈[0,1]
Wetting case: several open issues
I Integrated BM conditioned to stay non-negative is studied,but not its bridge
I Invariance principle seems in any case very difficult
I Entropic repulsion:
P(Z1 ≥ 0 , . . . , ZN ≥ 0
) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0
∣∣YN = 0,ZN = 0) ≈
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 30 / 34
The Models Free Energy Path Results Proofs Integrated random walk
The law of the excursions
Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2
t∈[0,1]
condit. on (YN ,ZN) = (0, 0) =⇒
It
t∈[0,1]
Wetting case: several open issues
I Integrated BM conditioned to stay non-negative is studied,but not its bridge
I Invariance principle seems in any case very difficult
I Entropic repulsion:
P(Z1 ≥ 0 , . . . , ZN ≥ 0
) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0
∣∣YN = 0,ZN = 0) ≈
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 30 / 34
The Models Free Energy Path Results Proofs Integrated random walk
The law of the excursions
Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2
t∈[0,1]
condit. on (YN ,ZN) = (0, 0) =⇒
It
t∈[0,1]
Wetting case: several open issues
I Integrated BM conditioned to stay non-negative is studied,but not its bridge
I Invariance principle seems in any case very difficult
I Entropic repulsion:
P(Z1 ≥ 0 , . . . , ZN ≥ 0
) ≈P(Z1 ≥ 0 , . . . , ZN ≥ 0
∣∣YN = 0,ZN = 0) ≈
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 30 / 34
The Models Free Energy Path Results Proofs Integrated random walk
The law of the excursions
Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2
t∈[0,1]
condit. on (YN ,ZN) = (0, 0) =⇒
It
t∈[0,1]
Wetting case: several open issues
I Integrated BM conditioned to stay non-negative is studied,but not its bridge
I Invariance principle seems in any case very difficult
I Entropic repulsion:
P(Z1 ≥ 0 , . . . , ZN ≥ 0
) ≈ ?
P(Z1 ≥ 0 , . . . , ZN ≥ 0
∣∣YN = 0,ZN = 0) ≈ ?
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 30 / 34
The Models Free Energy Path Results Proofs Integrated random walk
The law of the excursions
Pinning case: good control (Donsker’s inv. pr. + LLT)Z〈Nt〉N3/2
t∈[0,1]
condit. on (YN ,ZN) = (0, 0) =⇒
It
t∈[0,1]
Wetting case: several open issues
I Integrated BM conditioned to stay non-negative is studied,but not its bridge
I Invariance principle seems in any case very difficult
I Entropic repulsion:
P(Z1 ≥ 0 , . . . , ZN ≥ 0
) ≈ N−1/4 [Sinai (SRW)]
P(Z1 ≥ 0 , . . . , ZN ≥ 0
∣∣YN = 0,ZN = 0) ≈ N−1/2 [conj.]
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 30 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
Markov renewal processes
Given a (sub-)probability kernel Kx ,dy (n):∫y∈R
∑n∈N
Kx ,dy (n) = c ≤ 1 , ∀x ∈ R
we build the Markov renewal process τ with modulating chain J:
P(τi+1 − τi = n , Ji+1 ∈ dy
∣∣∣ Ji = x)
:= Kx ,dy (n)
The law of (τ, J) conditionally on N,N + 1 ⊆ τ is
P (τi = ti , Ji ∈ dyi
∣∣ N,N + 1 ⊆ τ) =1
CN
∏i
Kyi−1,dyi(ti − ti−1)
with CN = P(N,N + 1 ⊆ τ).
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 31 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
Markov renewal processes
Given a (sub-)probability kernel Kx ,dy (n):∫y∈R
∑n∈N
Kx ,dy (n) = c ≤ 1 , ∀x ∈ R
we build the Markov renewal process τ with modulating chain J:
P(τi+1 − τi = n , Ji+1 ∈ dy
∣∣∣ Ji = x)
:= Kx ,dy (n)
The law of (τ, J) conditionally on N,N + 1 ⊆ τ is
P (τi = ti , Ji ∈ dyi
∣∣ N,N + 1 ⊆ τ) =1
CN
∏i
Kyi−1,dyi(ti − ti−1)
with CN = P(N,N + 1 ⊆ τ).
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 31 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
The law of the contact set
Consider the following kernels: for n ∈ N and x , y ∈ R
Gpx ,dy (n) := ε
Px
(Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
Gwx ,dy (n) := ε
Px
(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
The law of (τ, J) is given by
Paε,N
(τi = ti , Ji ∈ dyi , i ≤ k
)=
1
Zaε,N
k∏i=1
Gayi−1,dyi
(ti − ti−1)
Reminds of Markov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 32 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
The law of the contact set
Consider the following kernels: for n ∈ N and x , y ∈ R
Gpx ,dy (n) := ε
Px
(Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
Gwx ,dy (n) := ε
Px
(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
The law of (τ, J) is given by
Paε,N
(τi = ti , Ji ∈ dyi , i ≤ k
)=
1
Zaε,N
k∏i=1
Gayi−1,dyi
(ti − ti−1)
Reminds of Markov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 32 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
The law of the contact set
Consider the following kernels: for n ∈ N and x , y ∈ R
Gpx ,dy (n) := ε
Px
(Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
Gwx ,dy (n) := ε
Px
(Zi ≥ 0 , i ≤ n , Zn−1 ∈ dy , Zn ∈ dz
)dz
∣∣∣∣z=0
The law of (τ, J) is given by
Paε,N
(τi = ti , Ji ∈ dyi , i ≤ k
)=
1
Zaε,N
k∏i=1
Gayi−1,dyi
(ti − ti−1)
Reminds of Markov renewal theory
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 32 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
Markov renewal processes
We exploit the invariance properties: for every F, v(y)
Paε,N
(τi = ti , Ji ∈ dyi , i ≤ k
)=
eFN
Zaε,N
k∏i=1
Gayi−1,dyi
(ti − ti−1) e−F(ti−ti−1) v(yi )
v(yi−1)
If we determine F, v(·) such that
Kx ,dy (n) := Gax ,dy (n) e−F·n
v(y)
v(x)
is a probability kernel, we have the crucial relation
Paε,N
(τi = ti , Ji ∈ dyi
)= P(τi = ti , Ji ∈ dyi
∣∣ N,N + 1 ⊆ τ)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 33 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
Markov renewal processes
We exploit the invariance properties: for every F, v(y)
Paε,N
(τi = ti , Ji ∈ dyi , i ≤ k
)=
eFN
Zaε,N
k∏i=1
Gayi−1,dyi
(ti − ti−1) e−F(ti−ti−1) v(yi )
v(yi−1)
If we determine F, v(·) such that
Kx ,dy (n) := Gax ,dy (n) e−F·n
v(y)
v(x)
is a probability kernel, we have the crucial relation
Paε,N
(τi = ti , Ji ∈ dyi
)= P(τi = ti , Ji ∈ dyi
∣∣ N,N + 1 ⊆ τ)Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 33 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
A Perron-Frobenius problem
It turns out that:
I F is the solution of the equation
spectral radius of
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
= 1
when such a solution exists and F = 0 otherwise.
I In fact F = Fa(ε) is the free energy
I v(·) is the principal eigenfunction:∫y∈R
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
v(y) = v(x)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 34 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
A Perron-Frobenius problem
It turns out that:
I F is the solution of the equation
spectral radius of
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
= 1
when such a solution exists and F = 0 otherwise.
I In fact F = Fa(ε) is the free energy
I v(·) is the principal eigenfunction:∫y∈R
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
v(y) = v(x)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 34 / 34
The Models Free Energy Path Results Proofs Markov renewal theory
A Perron-Frobenius problem
It turns out that:
I F is the solution of the equation
spectral radius of
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
= 1
when such a solution exists and F = 0 otherwise.
I In fact F = Fa(ε) is the free energy
I v(·) is the principal eigenfunction:∫y∈R
(∑n∈N
Gax ,dy (n) e−F·n
)x ,y
v(y) = v(x)
Francesco Caravenna (1+1)-Dimensional Fields with Laplacian Interaction April 30, 2008 34 / 34