Random Polymer Models
Disorder and Localization Phenomena
Giambattista Giacomin
University of Paris 7 – Denis Diderot
+ Probability Lab. (LPMA) Paris 6 & 7 CNRS U.M.A. 7599
A Polymer Close to a Membrane
exterior
more adhesive regions
interior
Does the polymer stick to the membrane?
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.1/65
A Polymer Close to a Membrane
exterior
more adhesive regions
interior
Does the polymer stick to the membrane?
For localization: contact energy (inhomogeneities?)
For delocalization: entropy (fluctuation freedom)
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.1/65
A Copolymer Near a Selective Interface
+ −→ hydrophobic (monomer)
− −→ hydrophilic
Contact Energy
Oil
−
−
−
+ + −
+
+
+
+
−
+ −
−
+
−
−
−
− + −
+
+
+
+
Interface
+
+
−
+
−
+
+
+
−
+
_
Water
−
Does the polymer localize at the interface?
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.2/65
A Copolymer Near a Selective Interface
+ −→ hydrophobic (monomer)
− −→ hydrophilic
Contact Energy
Oil
−
−
−
+ + −
+
+
+
+
−
+ −
−
+
−
−
−
− + −
+
+
+
+
Interface
+
+
−
+
−
+
+
+
−
+
_
Water
−
Does the polymer localize at the interface?
Possible relevant factors (in random order):
Distribution of charges (±)
Charge asymmetry (quantity or interaction)
Polymer intrinsic properties
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.2/65
DNA Unbinding (and More)
A-T
G-C
Loop
Loop
Reduced models (Peyrard-Bishop, Poland-Scheraga)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.3/65
DNA Unbinding (and More)
A-T
G-C
Loop
Loop
Reduced models (Peyrard-Bishop, Poland-Scheraga)
General framework:
pinning of chains on defect structures
The defect structure is (approximately) linear
It may (or may not) be possible to go around the defect
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.3/65
A General Mechanism...
On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read:
"In fact, there is a rather simple but general mathematical
mechanism which underlies a broad class of exactly
soluble one-dimensional models which display phase
transitions. This mechanism does not seem to be as well
appreciated as it merits and it operates in a number of
applications we wish to discuss."
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.4/65
A General Mechanism...
On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read:
"In fact, there is a rather simple but general mathematical
mechanism which underlies a broad class of exactly
soluble one-dimensional models which display phase
transitions. This mechanism does not seem to be as well
appreciated as it merits and it operates in a number of
applications we wish to discuss."
Fisher’s broad class of models is precisely the class we are
focusing on if we limit ourselves to homogeneous models (and
we start from there).
We aim at treating inhomogeneous (disordered) models.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.4/65
A General Mechanism...
On page 685 of Walks, Walls, Wetting, and Melting [M. E. Fisher, JSP (1984)] we read:
"In fact, there is a rather simple but general mathematical
mechanism which underlies a broad class of exactly
soluble one-dimensional models which display phase
transitions. This mechanism does not seem to be as well
appreciated as it merits and it operates in a number of
applications we wish to discuss."
G.G., Random Polymer Models, IC press (to appear)
From my web-page −→ research (.../pub/RPMay30.pdf.gz)
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.4/65
The Basic (Pinning) Model
Free process: Symmetric Random Walk {Sn }n
S0 = 0,
Sn =
n
X
Xj ,
j=1
{Xj }j IID with P (X1 = x) = 1/3, x ∈ {−1, 0, +1}.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.5/65
The Basic (Pinning) Model
Free process: Symmetric Random Walk {Sn }n
S0 = 0,
Sn =
n
X
Xj ,
j=1
{Xj }j IID with P (X1 = x) = 1/3, x ∈ {−1, 0, +1}.
Coupling parameter: β ∈ R
dPcN,β
dP
(S) =
1
c
ZN,β
exp β
N
X
n=1
!
1Sn =0 1SN =0 .
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.5/65
The Basic (Pinning) Model
Sn
0
N n
In this case: N = 16 and the energetic contribution is 6β .
dPcN,β
dP
(S) =
1
c
ZN,β
exp β
N
X
n=1
!
1Sn =0 1SN =0 .
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.6/65
The Basic Pinning Model
Notation and facts:
τ0 = 0 and τn = inf{m > τn−1 : Sm = 0} if τn−1 < ∞
(τn = ∞ otherwise).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.7/65
The Basic Pinning Model
Notation and facts:
τ0 = 0 and τn = inf{m > τn−1 : Sm = 0} if τn−1 < ∞
(τn = ∞ otherwise).
By the (strong) Markov property η := {ηi }i∈N ,
ηi = τi − τi−1 , is IID.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.7/65
The Basic Pinning Model
Notation and facts:
τ0 = 0 and τn = inf{m > τn−1 : Sm = 0} if τn−1 < ∞
(τn = ∞ otherwise).
By the (strong) Markov property η := {ηi }i∈N ,
ηi = τi − τi−1 , is IID.
K(n) := P(τ1 = n). Well known that
P
n→∞
−3/2
K(n) ∼ CK n
(CK > 0). Moreover n K(n) = 1
(i.e. S is recurrent).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.7/65
The Basic Pinning Model
Notation and facts:
τ0 = 0 and τn = inf{m > τn−1 : Sm = 0} if τn−1 < ∞
(τn = ∞ otherwise).
By the (strong) Markov property η := {ηi }i∈N ,
ηi = τi − τi−1 , is IID.
K(n) := P(τ1 = n). Well known that
P
n→∞
−3/2
K(n) ∼ CK n
(CK > 0). Moreover n K(n) = 1
(i.e. S is recurrent).
For us τ := {τn }n is also a random subset of N. So
expressions like N ∈ τ and τ ∩ A are meaningful.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.7/65
The Basic Pinning Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ 0, and
F (β) := 0 otherwise.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.8/65
The Basic Pinning Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ 0, and
F (β) := 0 otherwise.
Note that:
F (β)
> 0 for β > 0.
F : R → [0, ∞) is non decreasing and C 0 (in fact:
analytic on R \ {0}: βc = 0).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.8/65
The Basic Pinning Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ 0, and
F (β) := 0 otherwise.
Proposition 1. For every β
1
c
F (β) = lim
.
log ZN,β
N →∞ N
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.8/65
Proof of Prop. 1
The fundamental formula is
c
=
ZN,β
N
X
X
n
Y
exp(β)K(ℓj ).
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.9/65
Proof of Prop. 1
The fundamental formula is
c
=
ZN,β
N
X
X
n
Y
exp(β)K(ℓj ).
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
S
0
N
ℓ1
ℓ2
ℓ3
ℓ4
ℓ5
ℓ6
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.9/65
Proof of Prop. 1
The fundamental formula is
c
=
ZN,β
N
X
X
n
Y
exp(β)K(ℓj ).
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
e β (n) := exp(β)K(n) exp(−F(β)n). Then
Set K
c
ZN,β
= exp (F(β)N )
N
X
X
n
Y
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
e β (ℓj ).
K
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.9/65
Proof of Prop. 1
The fundamental formula is
c
=
ZN,β
N
X
X
n
Y
exp(β)K(ℓj ).
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
e β (n) := exp(β)K(n) exp(−F(β)n). Then
Set K
c
ZN,β
= exp (F(β)N )
N
X
X
n
Y
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
e β (ℓj ).
K
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.9/65
Proof of Prop. 1
e β the law of the renewal τ on N ∪ {∞} with
With P
e β (·):
inter-arrival law K
c
e β (N ∈ τ ),
ZN,β
= exp (F(β)N ) P
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.10/65
Proof of Prop. 1
e β the law of the renewal τ on N ∪ {∞} with
With P
e β (·):
inter-arrival law K
so
c
e β (N ∈ τ ),
ZN,β
= exp (F(β)N ) P
c
≤ exp (F(β)N ) ,
ZN,β
and it suffices to show that
1
e β (N ∈ τ ) ≥ 0.
lim inf log P
N →∞ N
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.10/65
Proof of Prop. 1
If β > βc : mKe β :=
P
e β (n) < ∞ and (by the
n
K
n
N →∞
e
Renewal Theorem) Pβ (N ∈ τ ) ∼ 1/mKe β > 0.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.11/65
Proof of Prop. 1
If β > βc : mKe β :=
P
e β (n) < ∞ and (by the
n
K
n
N →∞
e
Renewal Theorem) Pβ (N ∈ τ ) ∼ 1/mKe β > 0.
e β (N ∈ τ ) ≥ K
e β (N ) = exp(β)K(N ),
If β < βc : use P
which is bounded below by const./N 3/2 .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.11/65
Proof of Prop. 1
If β > βc : mKe β :=
P
e β (n) < ∞ and (by the
n
K
n
N →∞
e
Renewal Theorem) Pβ (N ∈ τ ) ∼ 1/mKe β > 0.
e β (N ∈ τ ) ≥ K
e β (N ) = exp(β)K(N ),
If β < βc : use P
which is bounded below by const./N 3/2 .
e β (·) = K(·) and
If β = βc : K
e β (N ∈ τ ) = P(N ∈ τ ) ≥ K(N ) ≥ const./N 3/2 .
P
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.11/65
Proof of Prop. 1
If β > βc : mKe β :=
P
e β (n) < ∞ and (by the
n
K
n
N →∞
e
Renewal Theorem) Pβ (N ∈ τ ) ∼ 1/mKe β > 0.
e β (N ∈ τ ) ≥ K
e β (N ) = exp(β)K(N ),
If β < βc : use P
which is bounded below by const./N 3/2 .
e β (·) = K(·) and
If β = βc : K
e β (N ∈ τ ) = P(N ∈ τ ) ≥ K(N ) ≥ const./N 3/2 .
P
The proof is complete.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.11/65
A Further Look at the Proof
We have in fact proven the sharp estimate
c
ZN,β
(1 + o(1))
exp (N F(β)) ,
=
mKe β
for every β .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.12/65
A Further Look at the Proof
We have in fact proven the sharp estimate
c
ZN,β
(1 + o(1))
exp (N F(β)) ,
=
mKe β
for every β .
This can be improved for β < βc
c
= (cβ + o(1))N −3/2 ,
ZN,β
and at criticality:
c
= (c0 + o(1))N −1/2 .
ZN,β
c
We’ll come back to this.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.12/65
What about the trajectories?
By convexity the contact fraction is:
1 c
1
c
∂β log ZN,β = lim
EN,β
∂β F(β) = lim
N →∞ N
N →∞ N
"
N
X
n=1
#
1Sn =0 ,
if ∂β F(β) exists.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.13/65
What about the trajectories?
By convexity the contact fraction is:
1 c
1
c
∂β log ZN,β = lim
EN,β
∂β F(β) = lim
N →∞ N
N →∞ N
"
N
X
#
1Sn =0 ,
n=1
if ∂β F(β) exists.
Therefore:
The contact fraction is zero for β < βc .
The contact fraction is positive for β > βc .
So the localization can be characterized by F(β) > 0.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.13/65
A basic (penetrable) wall model
Coupling parameter: h ∈ [0, ∞]
c,+
dPN,λ
dP
(S) =
1
c,+
ZN,h
exp −h
N
X
n=1
1Sn <0
!
1SN =0 .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.14/65
A basic (penetrable) wall model
Coupling parameter: h ∈ [0, ∞]
c,+
dPN,λ
dP
(S) =
1
c,+
ZN,h
exp −h
N
X
n=1
1Sn <0
!
1SN =0 .
One sees immediately that F(h) = 0 and that
#
"N
1 c,+ X
−∂h F(h) = lim
1Sn <0 ,
EN,h
N →∞ N
n=1
so the walk spends (essentially) all is time in the upper
half plane. Keyword: Entropic Repulsion.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.14/65
A basic (penetrable) wall model
For a closer look let us write
c,+
ZN,h
=
N
X
X
n
Y
1
n=1 P ℓ∈Nn :
j=1
n
j=1 ℓj =N
2
(exp(−h(ℓj − 1)) + 1) K(ℓj ).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.15/65
A basic (penetrable) wall model
For a closer look let us write
c,+
ZN,h
=
N
X
X
n=1 P ℓ∈Nn :
n
j=1 ℓj =N
n
Y
1
(exp(−h(ℓj − 1)) + 1) K(ℓj ).
2
{z
}
j=1 |
If we set Kh (ℓj ) and h > 0 we have
P
n Kh (n)
< 1.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.15/65
A basic (penetrable) wall model
For a closer look let us write
c,+
ZN,h
=
N
X
X
n=1 P ℓ∈Nn :
n
j=1 ℓj =N
n
Y
1
(exp(−h(ℓj − 1)) + 1) K(ℓj ).
2
{z
}
j=1 |
If we set Kh (ℓj ) and h > 0 we have
P
n Kh (n)
< 1.
c,+
converges to the
This strongly suggests that, as N → ∞, PN,h
terminating renewal τ with inter-arrival distribution Kh (·) and
results like (L large)
c,+
lim PN,h
(τ ∩ (L, N − L) = ∅) = 1 − oL (1).
N →∞
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.15/65
The General Homogeneous Model
If τ is a general renewal with inter-arrivals taking values
in N we introduce NN (τ ) = |τ ∩ {1, 2, . . . , N }| and
dPcN,β
dP
(τ ) =
1
c
ZN,β
exp (βNN (τ )) 1N ∈τ .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.16/65
The General Homogeneous Model
If τ is a general renewal with inter-arrivals taking values
in N we introduce NN (τ ) = |τ ∩ {1, 2, . . . , N }| and
dPcN,β
dP
(τ ) =
1
c
ZN,β
exp (βNN (τ )) 1N ∈τ .
We focus on inter-arrivals with law
L(n)
K(n) = 1+α ,
n
n = 1, 2, . . .
with L(·) slowly varying and α ≥ 0. We set ΣK :=
P
K(∞) := 1 − ΣK and mK := n∈N∪{∞} nK(n).
P
n∈N K(n),
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.16/65
The General Homogeneous Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ βc = − log ΣK ,
and F(β) := 0 otherwise.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.17/65
The General Homogeneous Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ βc = − log ΣK ,
and F(β) := 0 otherwise.
Therefore:
F (β)
> 0 for β > βc .
F : R → [0, ∞) is non decreasing and C 0 (in fact:
analytic on R \ {βc }).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.17/65
The General Homogeneous Model
Call F(β) the only solution of
X
K(n) exp(−F(β)n) = exp(−β),
n
when such a solution exists, that is for β ≥ βc = − log ΣK ,
and F(β) := 0 otherwise.
Proposition 2. For every β
1
c
F (β) = lim
.
log ZN,β
N →∞ N
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.17/65
The General Homogeneous model
Critical behavior:
Theorem 3. For every α ≥ 0 and every L(·) there exists a slowly
varying function L̂(·) such that for δ > 0
F (βc
+ δ) = δ 1/ min(1,α) L̂(1/δ).
P
< ∞ then
P
limδց0 L̂(1/δ) = ΣK / n∈N nK(n).
In particular if
n∈N nK(n)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.18/65
The General Homogeneous model
Critical behavior:
Theorem 3. For every α ≥ 0 and every L(·) there exists a slowly
varying function L̂(·) such that for δ > 0
F (βc
+ δ) = δ 1/ min(1,α) L̂(1/δ).
P
< ∞ then
P
limδց0 L̂(1/δ) = ΣK / n∈N nK(n).
In particular if
n∈N nK(n)
Therefore the transition is of k th order if α ∈ (1/k, 1/(k − 1)).
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.18/65
The General Homogeneous model
F (β)
N (β)
α>1
1
0
βc
1
β
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.19/65
The General Homogeneous Model
Sharp estimates on the partition function [Deuschel, G., Zambotti
PTRF2005], [Caravenna, G., Zambotti, EJP2006]:
Theorem 4. As N
→ ∞ we have:
1. (The localized regime.) If β
> βc then
c
∼
ZN,β
1
exp(F(β)N ).
mKe β
2. (The strictly delocalized regime.) If β
c
ZN,β
where ̺
< βc then
exp(β)
∼
K(N ),
2
(1 − ̺)
:= exp(β)ΣK (< 1).
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.20/65
The General Homogeneous model
Theorem 4 (Cont.)
3. (The critical regime. ) If we set K(n) :=
PN
P
j>n K(j) and
mN := n=0 K(n) then N 7→ mN is increasing and, if
α ≥ 1, slowly varying and
(
1/mN
if α ≥ 1,
c
ZN,βc ∼ ΣK ×
α sin(πα)N α−1 /(πL(N )) if α ∈ (0, 1).
If
P
n
nK(n) < ∞ then mN may be substituted by m∞ . If K(∞) = 0 then m∞ is the
mean of the inter-arrival.
Subtle result in the critical regime [R. A. Doney (1997)]. Open for α
= 0.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.21/65
The General Homogeneous model
Why are we so much interested in sharp estimates on
c ?
ZN,β
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.22/65
The General Homogeneous model
Why are we so much interested in sharp estimates on
c ?
ZN,β
(⇐=)
Sharp estimates =⇒ Sharp path behavior
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.22/65
The General Homogeneous model
Why are we so much interested in sharp estimates on
c ?
ZN,β
(⇐=)
Sharp estimates =⇒ Sharp path behavior
n
o
c
Proposition 5. For every β the sequence PN,β
converges
N
e β , the law of the renewal τeβ with inter-arrival K
e β (·).
weakly to P
τeβ is positive recurrent if β > βc .
τeβ is terminating if β < βc .
τeβ is recurrent if β = βc .
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.22/65
The General Homogeneous Model
Proof of Prop. 5: ℓ1 , ℓ2 ∈ N and q := PcN,β (τ1 = ℓ1 , τ2 = ℓ2 )
q =
c
eβ K(ℓ1 )eβ K(ℓ2 )ZN
−ℓ1 −ℓ2 ,β
c
ZN,β
with
RN := exp((ℓ1 + ℓ2 )F(β))
N →∞
e β (ℓ1 )K
e β (ℓ2 )RN ,
= K
c
ZN
−ℓ1 −ℓ2 ,β
c
ZN,β
and RN −→ 1 by the sharp estimates.
Scaling limits: τ (N ) :=
τ
N
e β (N − ℓ1 − ℓ2 ∈ τ )
P
,
=
e
Pβ (N ∈ τ )
∩ [0, 1]. Particularly interesting: critical case.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.23/65
Back to the "General Mechanism"
M. E. Fisher’s approach considers the Laplace transform
c :
of ZN,β
X
c
Θ(µ) :=
exp(−µN )ZN,β
,
N
which can be computed explicitly.
By super-additive arguments it is easy to show that
c
exists and this immediately implies
limN (1/N ) log ZN,β
that this limit coincides with inf {µ : Θ(µ) < ∞}.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.24/65
Back to the "General Mechanism"
M. E. Fisher’s approach considers the Laplace transform
c :
of ZN,β
X
c
Θ(µ) :=
exp(−µN )ZN,β
,
N
which can be computed explicitly.
By super-additive arguments it is easy to show that
c
exists and this immediately implies
limN (1/N ) log ZN,β
that this limit coincides with inf {µ : Θ(µ) < ∞}.
Extracting sharp estimates requires Tauberian arguments.
[Erdös, Pollard, Feller, Port, Garsia, Lamperti,...(1949 onwards)]
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.24/65
A Copolymer Near a Selective Interface
+ −→ hydrophobic (monomer)
− −→ hydrophilic
Contact Energy
Oil
−
−
−
+ + −
+
+
+
+
−
+ −
−
+
−
−
−
− + −
+
+
+
+
Interface
+
+
−
+
−
+
+
+
−
+
_
Water
−
Does the polymer localize at the interface?
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.25/65
A Copolymer Near a Selective Interface
+ −→ hydrophobic (monomer)
− −→ hydrophilic
Contact Energy
Oil
−
−
−
+ + −
+
+
+
+
−
+ −
−
+
−
−
−
− + −
+
+
+
+
Interface
+
+
−
+
−
+
+
+
−
+
_
Water
−
Does the polymer localize at the interface?
Possible relevant factors (in random order):
Distribution of charges (±)
Charge asymmetry (quantity or interaction)
Polymer intrinsic properties
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.25/65
Weakly Inhomogeneous Models
Periodic charge sequence ω := {ωn }n :
ωn ∈ R and ωn+T = ωn ,
for some T ∈ N and every n.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.26/65
Weakly Inhomogeneous Models
Periodic charge sequence ω := {ωn }n :
ωn ∈ R and ωn+T = ωn ,
for some T ∈ N and every n.
A copolymer model: S is the simple random walk (N ∈ 2N),
λ, h ≥ 0, ωn = (−1)n (so T = 2) and sign(0) = +1
!
N
c
X
dPN,ω,λ,h
1
(S) = c
exp λ
(ωn + h) sign (Sn ) 1SN =0 .
dP
ZN,ω,β
n=1
[Garel, Monthus, Orland (2000)], [Sommer, Daoud (1996)],...
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.26/65
Weakly Inhomogeneous Copolymer
Sn
+2
0
+2
0
−
+
−
−
+
−
+
0
λ
+
−
+
PN
−
+
−
n=1
0
+
−
(ωn + h) sign (Sn ) ,
sign(0) = +1
−
+
n
+
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.27/65
Weakly Inhomogeneous Copolymer
Observe that:
λ
N
X
(ωn + h) sign (Sn ) =
n=1
−2λ
N
X
(ωn + h) 1sign(Sn =−1) + λ
n=1
N
X
(ωn + h).
n=1
So: the copolymer is equivalent to a pinning model with
e :=
exp(β)
X 1 + exp (+2λ − 2λhn)
n
2
K(n),
e 1+exp(+2λ−2λhn) K(n).
e
and K(n) replaced by the probability K(n)
:= exp(−β)
2
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.28/65
Weakly Inhomogeneous Copolymer
From this computation we extract the phase diagram:
h
D
[GMO]-model
L
0
[SD]-model
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.29/65
Weakly Inhomogeneous Copolymer
More general case:
T > 2.
Periodic pinning and periodic copolymer interaction.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.30/65
Weakly Inhomogeneous Copolymer
More general case:
T > 2.
Periodic pinning and periodic copolymer interaction.
Possible strategy of solution: decimation (of step T ).
Problem: the model does not renormalize to a pinning.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.30/65
Weakly Inhomogeneous Copolymer
More general case:
T > 2.
Periodic pinning and periodic copolymer interaction.
Possible strategy of solution: decimation (of step T ).
Problem: the model does not renormalize to a pinning.
[Bolthausen, G. AAP2005]: an expression for the free energy (LD arguments).
[Caravenna, G., Zambotti2005,2006]: Renewal theory approach (sharp estimates and
path behavior). The underlying contact site process is a Markov renewal −→ richer
phenomenology.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.30/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
Free energy, localization and delocalization.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
Free energy, localization and delocalization.
Path properties (results and open problems).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
Free energy, localization and delocalization.
Path properties (results and open problems).
Free energy estimates (rare stretch strategy).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
Free energy, localization and delocalization.
Path properties (results and open problems).
Free energy estimates (rare stretch strategy).
A smoothing mechanism.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Strongly Inhomogeneous Models
Program:
Quenched disordered models: definitions.
Free energy, localization and delocalization.
Path properties (results and open problems).
Free energy estimates (rare stretch strategy).
A smoothing mechanism.
A number of questions can be treated for copolymers and
pinning at the same time. But on some other issues pinning and
copolymer models have sharply different behaviors.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.31/65
Disordered Models: Definition
Free process: (τ, s) couple of independent processes s.t.
τ renewal process on N ∪ {∞} with inter-arrival law
K(·) (K(n) = L(n)/n1+α );
s = {sj }j∈N IID with P(s1 = ±1) = 1/2;
S: walk s.t. {n ∈ N ∪ {0} : Sn = 0} = τ \ {∞} and
sign(Sn ) = sτj , j = inf{i : τi ≥ n}.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.32/65
Disordered Models: Definition
Free process: (τ, s) couple of independent processes s.t.
τ renewal process on N ∪ {∞} with inter-arrival law
K(·) (K(n) = L(n)/n1+α );
s = {sj }j∈N IID with P(s1 = ±1) = 1/2;
S: walk s.t. {n ∈ N ∪ {0} : Sn = 0} = τ \ {∞} and
sign(Sn ) = sτj , j = inf{i : τi ≥ n}.
Parameters: v
Charges ω ∈ RN (centered quenched disorder)
dPcN,ω,v
dP
(S) =
1
c
ZN,ω,v
v
exp HN,ω (S) 1{SN =0} .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.32/65
Disordered Models: Definition
Free process: (τ, s) couple of independent processes s.t.
τ renewal process on N ∪ {∞} with inter-arrival law
K(·) (K(n) = L(n)/n1+α );
s = {sj }j∈N IID with P(s1 = ±1) = 1/2;
S: walk s.t. {n ∈ N ∪ {0} : Sn = 0} = τ \ {∞} and
sign(Sn ) = sτj , j = inf{i : τi ≥ n}.
Parameters: v
Charges ω ∈ RN (centered quenched disorder)
dPfN,ω,v
dP
(S) =
1
c
ZN,ω,v
exp
v
HN,ω (S)
.1{SN =0}
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.32/65
Disordered Models: Definition
Copolymer: v = (λ, h), λ, h ≥ 0
λ,h
HN,ω
(S) = λ
N
X
(ωn + h) sign (Sn )
n=1
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.33/65
Disordered Models: Definition
Copolymer: v = (λ, h), λ, h ≥ 0
λ,h
HN,ω
(S) = λ
N
X
n=1
(ωn + h) sign (Sn ) = λ
NN (τ )
X
j=1
sj
τj
X
(ωn + h) .
n=τj−1 +1
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.33/65
Disordered Models: Definition
Copolymer: v = (λ, h), λ, h ≥ 0
λ,h
HN,ω
(S) = λ
N
X
(ωn + h) sign (Sn ) = λ
n=1
NN (τ )
X
j=1
sj
τj
X
(ωn + h) .
n=τj−1 +1
Pinning at a defect line: v = (β, h), β ≥ 0, h ∈ R
β,h
HN,ω
(S) =
N
X
n=1
(βωn − h) 1{Sn =0}
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.33/65
Disordered Models: Definition
Copolymer: v = (λ, h), λ, h ≥ 0
λ,h
HN,ω
(S) = λ
N
X
(ωn + h) sign (Sn ) = λ
n=1
NN (τ )
X
sj
j=1
τj
X
(ωn + h) .
n=τj−1 +1
Pinning at a defect line: v = (β, h), β ≥ 0, h ∈ R
β,h
HN,ω
(S)
=
N
X
n=1
(βωn − h) 1{Sn =0} =
X
(βωn − h) .
n∈τ :
1≤n≤N
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.33/65
Choice of the disorder
Random (quenched):
{ωn }n IID (law P) such that
M(t) := E[exp(tω1 )] < ∞ for |t| small
E [ω1 ] = 0 (let us say ω1 ∼ −ω1 )
2
Without loss of generality: E ω1 = 1.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.34/65
Choice of the disorder
Random (quenched):
{ωn }n IID (law P) such that
M(t) := E[exp(tω1 )] < ∞ for |t| small
E [ω1 ] = 0 (let us say ω1 ∼ −ω1 )
2
Without loss of generality: E ω1 = 1.
Copolymer: λ, h ≥ 0, ∆n := (1 − sign(Sn ))/2 ∈ {0, 1}
λ,h
HN,ω
(S) = −2λ
N
X
(ωn + h) ∆n + cN (ω)
n=1
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.34/65
The Free Energy
Theorem 5. The limit of the sequence of r.v.’s
n
1
N
a
log ZN,ω,v
o
N
exists P( dω)–a.s. and in L1 (P), it is P( dω)–a.s. equal to a constant
(self-averaging!) that we denote by F(v).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.35/65
The Free Energy
Theorem 5. The limit of the sequence of r.v.’s
n
1
N
a
log ZN,ω,v
o
N
exists P( dω)–a.s. and in L1 (P), it is P( dω)–a.s. equal to a constant
(self-averaging!) that we denote by F(v).
Several "different" proofs: for example observe that
c
c
c
log ZN,ω
≥ log ZM,ω
+ log ZN
−M,θM ω
o
n
c
for M = 0, . . . , N . So E log ZN,ω
N
is superadditive, so
1
1
c
c
E log ZN,ω = sup E log ZN,ω
.
lim
N →∞ N
N N
A concentration bound suffices to conclude.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.35/65
Free energy approach
The free energy:
h
i
1
1
λ,h
f
log ZN,ω,v
log E exp HN,ω
(S) = F(λ, h)
lim
= lim
N →∞ N
N →∞ N
P( dω)–a.s. and in L1 (P).
Identification of the free energy of the delocalized state:
”i
h
“
λ,h
E exp HN,ω (S) ≥
"
E exp
−2λ
N
X
(ωn + h)∆n
n=1
!
; Sn > 0 for n = 1, 2, . . . , N
#
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.36/65
Free energy approach
The free energy:
h
i
1
1
λ,h
f
log ZN,ω,v
log E exp HN,ω
(S) = F(λ, h)
lim
= lim
N →∞ N
N →∞ N
P( dω)–a.s. and in L1 (P).
Identification of the free energy of the delocalized state:
”i
h
“
λ,h
E exp HN,ω (S) ≥
E
"
P ( · ) ≈ 1/N α
#
z
}|
{
1 ; Sn > 0 for n = 1, 2, . . . , N = exp(o(N ))
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.36/65
Free energy approach
The free energy:
h
i
1
1
λ,h
f
log ZN,ω,v
log E exp HN,ω
(S) = F(λ, h) ≥ 0
lim
= lim
N →∞ N
N →∞ N
P( dω)–a.s. and in L1 (P).
Identification of the free energy of the delocalized state:
”i
h
“
λ,h
E exp HN,ω (S) ≥
E
"
P ( · ) ≈ 1/N α
#
}|
{
z
1 ; Sn > 0 for n = 1, 2, . . . , N = exp(o(N ))
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.36/65
Localized and Delocalized regions
D = {v : F(v) = 0}
L = {v : F(v) > 0}
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.37/65
Localized and Delocalized regions
D = {v : F(v) = 0}
L = {v : F(v) > 0}
The analysis (not obvious!):
Understanding the phase diagram
What does v ∈ L (or D) mean for the polymer
trajectories?
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.37/65
Localized and Delocalized regions
D = {v : F(v) = 0}
L = {v : F(v) > 0}
The analysis (not obvious!):
Understanding the phase diagram
What does v ∈ L (or D) mean for the polymer
trajectories?
Chronological math approach:
Path analysis −→ free energy approach
[S]:=[Sinai TPA93] −→ [BdH97]:=[Bolthausen, den Hollander AP97]
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.37/65
Pinning: Phase Diagram
h
Energy=
P
n∈τ ∩[1,N ] (βωn
− h)
D
hc (0) + log M(·)
hc (·)
β
0
hc (0) + hLB (·)
hc (0)
L
There exists hc (·) (convex, increasing) separating L and D.
The upper bound is the annealed bound.
hLB (·) ≥ 0 is easy. hLB (·) > 0: [Alexander, Sidoravicius 05], [G]
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.38/65
Copolymer: Phase Diagram
[S], [BdH], [BG]:=[Bodineau and G. JSP04], [GT05]:=[G., Toninelli PTRF05]
hc (λ)
D
L
0
Energy= λ
λ
PN
n=1 (ωn
+ h)1sign(Sn )=−1
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.39/65
Copolymer: Phase Diagram
[S], [BdH], [BG]:=[Bodineau and G. JSP04], [GT05]:=[G., Toninelli PTRF05]
P
( n K(n) = 1)
hc (λ)
D
L
0
λ
√
As λ ց 0: hc (λ) = mc λ + o(λ), mc ∈ [1/(1 + α), 1]. Recently: mc ≥ 1/ 1 + α if
α ≥ 1.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.39/65
Phase diagram estimates
h(m) (λ) :=
h
1
log M (2mλ) ,
2mλ
h′c (0) ∈ [1/(1 + α), 1]
h(1/(1+α)) (λ) ≤ hc (λ) ≤ h(1) (λ)
h(1) (λ)
hc (λ)
h(1/(1+α)) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.40/65
Phase diagram estimates
n→∞
The physical literature is split: α = 1/2 and L(n) ∼ c > 0
For h(1) (·) = hc (·): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case)
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.41/65
Phase diagram estimates
n→∞
The physical literature is split: α = 1/2 and L(n) ∼ c > 0
For h(1) (·) = hc (·): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case)
For h(2/3) (·) = hc (·):
[Stepanov et al. 98] (Replica, Gauss)
[Monthus 00] (Fisher’s renormalization scheme)
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.41/65
What Does the Computer Say?
[Caravenna, G., Gubinelli JSP06]
Statistical test based on concentration and super-additivity: hc (·) > h(2/3) (·)
h′c (0) ≈ 0.83 ± 0.01
[Garel, Monthus 05]
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.42/65
Testing for Localization
Basic ingredients:
n
c
Super-additivity of E log ZN,ω,v
o
N
implies:
c
> 0.
v ∈ L ⇐⇒ there exists N s.t. E log ZN,ω,v
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.43/65
Testing for Localization
Basic ingredients:
n
c
Super-additivity of E log ZN,ω,v
o
N
implies:
c
> 0.
v ∈ L ⇐⇒ there exists N s.t. E log ZN,ω,v
For a wide class of r.v.’s ω1 we have the concentration
bound: there exist c1 , c2 > 0 such that for every convex
G : Rk → R satisfying |G(r) − G(r′ )| ≤ C|r − r′ | we have
that for every t
2
P (|G(ω) − EG(ω)| ≥ t) ≤ c1 exp −c2 t /C
2
.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.43/65
Testing for Localization
In our case k = nN and G(ω) :=
0
1
P@
n
n
X
j=1
c
log ZN,θ
j−1 ω
1
n
Pn
j=1
2
2
c
log ZN,θ
j−1 ω . Now C = 16λ N/n and
1
i
h
`
´
c
2
2
A
− E log ZN,ω ≥ t ≤ c1 exp −c2 nt /(16λ N ) .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.44/65
Testing for Localization
In our case k = nN and G(ω) :=
0
1
P@
n
n
X
j=1
c
log ZN,θ
j−1 ω
1
n
Pn
j=1
2
2
c
log ZN,θ
j−1 ω . Now C = 16λ N/n and
1
i
h
`
´
c
2
2
A
− E log ZN,ω ≥ t ≤ c1 exp −c2 nt /(16λ N ) .
Therefore:
c ] ≤ 0.
fix N and make the hypothesis that E[log ZN,ω
c
Compute n independent samples of log ZN,ω
(comp. time
O(nN 3/2 )) and compute the mean û of the results.
If û > 0 we refuse the hypothesis (therefore: localization!)
2
2
with a level of error smaller than c1 exp −c2 nû /(16λ N ) .
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.44/65
A quick look at the paths
Real question: what do the paths do?
The localized phase ∼ Large Deviations regime
[S], [Albeverio, Zhou JSP96]
[Biskup, den Hollander AAP99] ←− Gibbs measure viewpoint
[G., Toninelli Alea06] ←− two distinct correlation lengths
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.45/65
A quick look at the paths
Real question: what do the paths do?
The localized phase ∼ Large Deviations regime
[S], [Albeverio, Zhou JSP96]
[Biskup, den Hollander AAP99] ←− Gibbs measure viewpoint
[G., Toninelli Alea06] ←− two distinct correlation lengths
The delocalized phase ∼ Moderate Deviations
◦
If v ∈D =⇒ zero density (o(N )) of visits to the interface
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.45/65
A quick look at the paths
Real question: what do the paths do?
The localized phase ∼ Large Deviations regime
[S], [Albeverio, Zhou JSP96]
[Biskup, den Hollander AAP99] ←− Gibbs measure viewpoint
[G., Toninelli Alea06] ←− two distinct correlation lengths
The delocalized phase ∼ Moderate Deviations
◦
If v ∈D =⇒ zero density (o(N )) of visits to the interface
[G., Toninelli PTRF05]: O(log N ) and O(1) results
If v ∈ ∂D: O(N 2/3 ) visits to the interface [Toninelli JSP06]
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.45/65
A quick look at the paths
Real question: what do the paths do?
The localized phase ∼ Large Deviations regime
[S], [Albeverio, Zhou JSP96]
[Biskup, den Hollander AAP99] ←− Gibbs measure viewpoint
[G., Toninelli Alea06] ←− two distinct correlation lengths
The delocalized phase ∼ Moderate Deviations
◦
If v ∈D =⇒ zero density (o(N )) of visits to the interface
[G., Toninelli PTRF05]: O(log N ) and O(1) results
If v ∈ ∂D: O(N 2/3 ) visits to the interface [Toninelli JSP06]
◦
Expected in D (?): scaling toward Brownian meander
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.45/65
On the Delocalized phase [GT05]
Theorem 6. Besides the standard assumptions on ω , let us assume that there
exist positive constants c1 and c2 s.t.
2
P (G(ω1 , . . . , ωk ) − EG(ω1 , . . . , ωk ) > t) ≤ c1 exp −c2 t
for every t
≥ 0 and every (convex) G : Rk → R such that
,
◦
|G(x) − G(y)| ≤ |x − y|. Then for every (β, h) ∈D there exist two
positive constants c and q such that
EPaN,ω (NN (τ ) ≥ n) ≤ exp (−cn) ,
both for a
= c and a = f and for every N and n ≥ q log N .
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.46/65
Proof of Theorem 6
Set
1
a
log ZN,ω
(NN (τ ) = n) ,
:=
N
a .
and FaN,ω (β, h) := (1/N ) log ZN,ω
a
F N,ω (β, h; n)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.47/65
Proof of Theorem 6
Set
1
a
log ZN,ω
(NN (τ ) = n) ,
:=
N
a .
and FaN,ω (β, h) := (1/N ) log ZN,ω
a
F N,ω (β, h; n)
Important because:
PcN,ω
(NN (τ ) ≥ m) =
1
c
ZN,ω
X
c
ZN,ω
(NN (τ ) = n) .
n≥m
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.47/65
Proof of Theorem 6
Set
1
a
log ZN,ω
(NN (τ ) = n) ,
:=
N
a .
and FaN,ω (β, h) := (1/N ) log ZN,ω
We have:
a
F N,ω (β, h; n)
Lemma 7. Same assumptions as in Th. 6. We have
2 2
c2 u N
a
a
,
P FN,ω (β, h; n) − E FN,ω (β, h; n) ≥ u ≤ c1 exp −
2
β n
for every N
∈ N, n ∈ {0, 1, . . . , N } and every u ≥ 0.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.47/65
Proof of Theorem 6
Set
1
a
log ZN,ω
(NN (τ ) = n) ,
:=
N
a .
and FaN,ω (β, h) := (1/N ) log ZN,ω
We have:
a
F N,ω (β, h; n)
Lemma 7. Same assumptions as in Th. 6. We have
2 2
c2 u N
a
a
,
P FN,ω (β, h; n) − E FN,ω (β, h; n) ≥ u ≤ c1 exp −
2
β n
for every N
∈ N, n ∈ {0, 1, . . . , N } and every u ≥ 0.
If n = N we recover the standard fluctuation/concentration bound, but for small n the
bound improves substantially.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.47/65
Proof of Theorem 6
◦
Choose (β, h) ∈D and ε > 0 s.t. (β, h − ε) ∈ D.
Super-additivity guarantees:
(β, h) ∈ D ⇐⇒ E
c
FN,ω (β, h)
≤ 0 for every N ∈ N.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.48/65
Proof of Theorem 6
◦
Choose (β, h) ∈D and ε > 0 s.t. (β, h − ε) ∈ D.
Super-additivity guarantees:
(β, h) ∈ D ⇐⇒ E
c
FN,ω (β, h)
≤ 0 for every N ∈ N.
For every ω we have the equivalence
c
FN,ω (β, h; n)
and
1
≥ − εn/N ⇐⇒
2
FcN,ω (β, h − ε; n)
≤
c
F N,ω (β, h
1
− ε; n) ≥ εn/N,
2
F cN,ω (β, h − ε).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.48/65
Proof of Theorem 6
◦
Choose (β, h) ∈D and ε > 0 s.t. (β, h − ε) ∈ D.
Super-additivity guarantees:
(β, h) ∈ D ⇐⇒ E
c
FN,ω (β, h)
≤ 0 for every N ∈ N.
For every ω we have the equivalence
c
FN,ω (β, h; n)
and
1
≥ − εn/N ⇐⇒
2
FcN,ω (β, h − ε; n)
≤
c
F N,ω (β, h
1
− ε; n) ≥ εn/N,
2
F cN,ω (β, h − ε).
Note that EFcN,ω (β, h − ε) ≤ 0.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.48/65
Proof of Theorem 6
Therefore (Lemma 7)
«
1
P FcN,ω (β, h; n) ≥ − εn/N ≤
2
„
«
h
i
1
P FcN,ω (β, h − ε; n) − E FcN,ω (β, h − ε; n) ≥ εn/N
2
„
«2 !
ε
≤ c1 exp −c2 n
.
2|β|
„
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.49/65
Proof of Theorem 6
Therefore (Lemma 7)
«
1
P FcN,ω (β, h; n) ≥ − εn/N ≤
2
„
«
h
i
1
P FcN,ω (β, h − ε; n) − E FcN,ω (β, h − ε; n) ≥ εn/N
2
„
«2 !
ε
≤ c1 exp −c2 n
.
2|β|
„
With Em := {ω : ∃n ≥ m s.t.
X
„
F cN,ω (β, h; n)
≥ −εn/2N }, we have
1
P (Em ) ≤
P FcN,ω (β, h; n) ≥ − εn/N
2
n≥m
«
≤ C1−1 exp(−C1 m),
for every m and some C1 > 0.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.49/65
Proof of Theorem 6
∁ . We have (α > α+ ):
Choose now ω ∈ Em
PcN,ω (NN (τ ) ≥ m) =
≤ C2 N
1+α+
1
c
ZN,ω
X
c
ZN,ω
(NN (τ ) = n)
n≥m
exp (−βωN + h)
X
exp (−εn/2)
n≥m
≤ C3 N 1+α+ exp (−βωN + h) exp (−εm/2) ,
and by taking the expectation we conclude.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.50/65
Localization and Smoothing
Program:
Copolymer model: lower bound on critical curve.
All models: smoothing inequality.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.51/65
Localization and Smoothing
Program:
Copolymer model: lower bound on critical curve.
All models: smoothing inequality.
Common idea: role of rare stretches of charges
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.51/65
Phase diagram estimates
h(m) (λ) :=
h
1
log M (2mλ) ,
2mλ
h′c (0) ∈ [1/(1 + α), 1]
h(1/(1+α)) (λ) ≤ hc (λ) ≤ h(1) (λ)
h(1) (λ)
hc (λ)
h(1/(1+α)) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.52/65
Phase diagram estimates
n→∞
The physical literature is split: α = 1/2 and L(n) ∼ c > 0
For h(1) (·) = hc (·): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case)
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.53/65
Phase diagram estimates
n→∞
The physical literature is split: α = 1/2 and L(n) ∼ c > 0
For h(1) (·) = hc (·): [Garel et al. 89], [Maritan, Trovato 99] (Replica, Gauss case)
For h(2/3) (·) = hc (·):
[Stepanov et al. 98] (Replica, Gauss)
[Monthus 00] (Fisher’s renormalization scheme)
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.53/65
What Does the Computer Say?
[Caravenna, G., Gubinelli JSP06]
Statistical test based on concentration and super-additivity: hc (·) > h(2/3) (·)
h′c (0) ≈ 0.83 ± 0.01
[Garel, Monthus 05]
h
h(1) (λ)
hc (λ)
h(2/3) (λ)
0
λ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.54/65
Rare Stretch Strategy
We want to prove:
h
(m)
1
(λ) :=
log M (2mλ) ,
2mλ
h(1/(1+α)) (λ) ≤ hc (λ)
(BG04)
Picture to keep in mind: N → ∞ and then ℓ → ∞
Sn
Atypical ωn ’s
N n
0
ℓ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.55/65
Proof of inequality (BG04)
Sn
Qj ≤ −qℓ
Q1 Q2
Nn
0
ℓ
G1 ℓ/p(ℓ)
with Qj :=
Pjℓ
G2 ℓ/p(ℓ)
n=(j−1)ℓ+1 (ωj
ℓ
+ h). Note that {Qj }j is IID and
ℓ→∞
p(ℓ) := P (Q1 ≤ −qℓ) ≍ exp (−ℓΣ(−q − h)) ,
with Σ(·) the Cramér LD functional. {Gj }j IID too.
We restrict ZN,ω to the trajectories on the figure (lower bound).
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.56/65
Proof of inequality (BG04)
In one excursion the contribution to log ZN,ω is:
ℓ
+log (K(ℓ) exp(2λqℓ)) ≈ −(1+α)Σ(−q−h)ℓ+2λqℓ.
log K Gj
p(ℓ)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.57/65
Proof of inequality (BG04)
In one excursion the contribution to log ZN,ω is:
ℓ
+log (K(ℓ) exp(2λqℓ)) ≈ −(1+α)Σ(−q−h)ℓ+2λqℓ.
log K Gj
p(ℓ)
We can still optimize over q: we gain if
0 < gain := sup (2λq − (1 + α)Σ(−q − h))
q
= −2λh + (1 + α) sup ((−2λ/(1 + α))q − Σ(q))
q
= −2λh + (1 + α) log M (−2λ/(1 + α)) ,
obtaining, for ℓ sufficiently large, log ZN,ω ≥ N p(ℓ)gain/2.
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.57/65
Proof of inequality (BG04)
In one excursion the contribution to log ZN,ω is:
ℓ
+log (K(ℓ) exp(2λqℓ)) ≈ −(1+α)Σ(−q−h)ℓ+2λqℓ.
log K Gj
p(ℓ)
We can still optimize over q: we gain if
0 < gain := sup (2λq − (1 + α)Σ(−q − h))
q
= −2λh + (1 + α) sup ((−2λ/(1 + α))q − Σ(q))
q
= −2λh + (1 + α) log M (−2λ/(1 + α)) ,
obtaining, for ℓ sufficiently large, log ZN,ω ≥ N p(ℓ)gain/2.
Since gain> 0 ⇐⇒ h < h1/(1+α) (λ) we are done.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.57/65
On the Regularity of the Free Energy
Theorem 8. [G. and Toninelli Alea06] F(·) ∈ C ∞ in L.
The proof is based on estimates on decay of correlations of local
observables and on [von Dreifus, Klein and Fernando Perez CMP95]
Intriguing issues: Griffiths singularities?
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.58/65
Regularity at criticality
Stated for pinning (β → λ for copolymers)
Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of
disorders (including ω1 bounded and ω1
∼ N (0, 1)), for every β > 0 there
exists C such that for every h
F(β, h)
≤ (1 + α)C (h − hc (β))2 .
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.59/65
Regularity at criticality
Stated for pinning (β → λ for copolymers)
Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of
disorders (including ω1 bounded and ω1
∼ N (0, 1)), for every β > 0 there
exists C such that for every h
F(β, h)
≤ (1 + α)C (h − hc (β))2 .
Possibly more transparent when written as
0 ≤ F(β, h) − F(β, hc (β)) ≤ (1 + α)C (h − hc (β))2 ,
(the result is non trivial only for h < hc (β).)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.59/65
Regularity at criticality
Stated for pinning (β → λ for copolymers)
Theorem 9. [G. and Toninelli, CMP06 and PRL06]. For a wide class of
disorders (including ω1 bounded and ω1
∼ N (0, 1)), for every β > 0 there
exists C such that for every h
F(β, h)
≤ (1 + α)C (h − hc (β))2 .
Rephrasing once more:
F (λ, ·) is C 1,1 at hc (β) =⇒ the transition is at least of
second order (almost third...)
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.59/65
Smoothing?
Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90]
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.60/65
Smoothing?
Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90]
We need to precise:
the pure (i.e. non disordered) model
the regularity of F for the pure model
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.60/65
Smoothing?
Is this a smoothing/rounding effect? [Aizenman, Wehr CMP 90]
We need to precise:
the pure (i.e. non disordered) model
the regularity of F for the pure model
The pure models are:
Pinning: set β = 0 −→ just reward −h per visit.
Explicitly solvable (seen).
Copolymer: periodic (T ) distribution of charges.
(Almost) explicitly solvable too.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.60/65
Smoothing! (Sometimes)
In all cases:
F
pure
hրhc
(h) ∼ (hc − h)1/ min(α,1) L̂(hc − h),
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.61/65
Smoothing! (Sometimes)
In all cases:
F
pure
hրhc
(h) ∼ (hc − h)1/ min(α,1) L̂(hc − h),
Disordered above / pure below
Smoothing: like α = 1/2
0 ...
...
RW model
1/4 1/3
α = 1/2
second order
4th 3rd order
PS model
α=1
first order
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.61/65
Smoothing! (Sometimes)
In all cases:
F
pure
hրhc
(h) ∼ (hc − h)1/ min(α,1) L̂(hc − h),
Disordered above / pure below
Smoothing: like α = 1/2
0 ...
...
RW model
1/4 1/3
α = 1/2
second order
4th 3rd order
PS model
α=1
first order
Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.61/65
Smoothing! (Sometimes)
In all cases:
F
pure
hրhc
(h) ∼ (hc − h)1/ min(α,1) L̂(hc − h),
Disordered above / pure below
Smoothing: like α = 1/2
0 ...
...
RW model
1/4 1/3
α = 1/2
second order
4th 3rd order
PS model
α=1
first order
Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2)
[Forgacs et al. 1986], [Derrida et al. 1992] ... (!)
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.61/65
Smoothing! (Sometimes)
In all cases:
F
pure
hրhc
(h) ∼ (hc − h)1/ min(α,1) L̂(hc − h),
Disordered above / pure below
Smoothing: like α = 1/2
0 ...
...
RW model
1/4 1/3
α = 1/2
second order
4th 3rd order
PS model
α=1
first order
Harris (pinning): noise is relevant if α > 1/2 (irrelevant if α < 1/2)
[Cule, Hwa 97], [Tang,Chaté 01], [Garel, Monthus 05], [Coluzzi 05]
(Biomath: DNA denaturation, Poland-Scheraga model)
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.61/65
Rare stretch strategy (again)
On the proof of:
F (λ, h)
≤ (1 + α)C(β) (h − hc (β))2
(GT06)
Picture to keep in mind: N → ∞ and then ℓ → ∞
Sn
Atypical ωn ’s
N n
0
ℓ
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.62/65
On the proof of the smoothing inequality (Th. 9)
Restricted set-up: ω1 ∼ N (0, 1)
and only pinning model (energy=
P
n∈τ ∩[1,N ] (βωn
− h)).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.63/65
On the proof of the smoothing inequality (Th. 9)
Restricted set-up: ω1 ∼ N (0, 1)
and only pinning model (energy=
1
ℓ
P
Pℓ
n∈τ ∩[1,N ] (βωn
− h)).
Note that, observing
n=1 ωn ≥ δ/β implies for ℓ large
(roughly) observing ωj ∼ N (δ/β, 1) for j = 1, . . . , ℓ and therefore
log Zℓ,ω,β,h ≈ ℓF(β, h − δ).
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.63/65
On the proof of the smoothing inequality (Th. 9)
Restricted set-up: ω1 ∼ N (0, 1)
and only pinning model (energy=
1
ℓ
P
n∈τ ∩[1,N ] (βωn
Pℓ
− h)).
Note that, observing
n=1 ωn ≥ δ/β implies for ℓ large
(roughly) observing ωj ∼ N (δ/β, 1) for j = 1, . . . , ℓ and therefore
log Zℓ,ω,β,h ≈ ℓF(β, h − δ).
Precisely:
1
log P (log Zℓ,ω,β,h
ℓ→∞ ℓ
lim
2
1 δ
≥ ℓF(β, h − δ)/2) ≥ −
.
2 β
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.63/65
On the proof of the smoothing inequality (Th. 9)
Now apply the rare stretch strategy:
F (β, h)
≥ p(ℓ)
!
2
δ
1
F(β, h − δ) − 2(1 + α)
+ oℓ (1) .
2
β
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.64/65
On the proof of the smoothing inequality (Th. 9)
Now apply the rare stretch strategy:
F (β, h)
≥ p(ℓ)
!
2
δ
1
F(β, h − δ) − 2(1 + α)
+ oℓ (1) .
2
β
Set h = hc (β), so that the LHS is zero and we obtain that for
every ℓ
2
1
δ
0 ≥ F(β, h − δ) − 2(1 + α)
+ oℓ (1).
2
β
▽Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.64/65
On the proof of the smoothing inequality (Th. 9)
Now apply the rare stretch strategy:
F (β, h)
≥ p(ℓ)
!
2
δ
1
F(β, h − δ) − 2(1 + α)
+ oℓ (1) .
2
β
Set h = hc (β), so that the LHS is zero and we obtain that for
every ℓ
2
1
δ
0 ≥ F(β, h − δ) − 2(1 + α)
+ oℓ (1).
2
β
By letting ℓ → ∞ we obtain the smoothing inequality.
Obs.: no CLT and no boundary! Rather: LDs and flexibility.
Stochastic Processes in Mathematical Physics, Firenze June 19–23, 2006 – p.64/65
THE END
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