Continuous interfaces with disorder:
Even strong pinning is too weak in 2 dimensions
Christof Külske ∗and Enza Orlandi†
November 4, 2018
Abstract
We consider statistical mechanics models of continuous height effective interfaces
in the presence of a delta-pinning of strength ε at height zero. There is a detailed
mathematical understanding of the depinning transition in 2 dimensions without
disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays
bounded uniformly in the volume for ε > 0 and diverges like | log ε| for ε ↓ 0
How does the presence of a quenched disorder term in the Hamiltonian modify this
transition?
We show that an arbitarily weak random field term is enough to beat an ar-
bitrarily strong delta-pinning in 2 dimensions and will cause delocalization. The
proof is based on a rigorous lower bound for the overlap between local magnetiza-
tions and random fields in finite volume. In 2 dimensions it implies growth faster
than the volume which is a contradiction to localization. We also derive a simple
complementary inequality which shows that in higher dimensions the fraction of
pinned sites converges to one with ε ↑ ∞.
AMS 2000 subject classification: 60K57, 82B24,82B44.
1 Introduction
1.1 The setup
The study of lattice effective interface models, continous and discrete, has a long tradi-
tion in statistical mechanics [14, 5, 9, 10, 13, 2, 3, 4].
The model we study is given in terms of variables ϕi ∈ R which, physically speaking,
are thought to represent height variables of a random surface at the sites i ∈ Zd.
Mathematically speaking they are just continuous unbounded (spin) variables. The
model is defined in terms of: a pair potential V , a quenched random term, and a
pinning term at interface height zero.
More precisely, we are interested in the behavior of the quenched finite-volume Gibbs
measures in a finite volume Λ⊂Zd with fixed boundary condition at height zero, given
University of Groningen, Department of Mathematics and Computing Sciences, Blauwborgje 3,
9747 AC Groningen, The Netherlands kuelske@math.rug.nl, http://www.math.rug.nl/∼kuelske/
Dipartimento di Matematica, Universit degli Studi ”Roma Tre”, Largo San Leonardo Murialdo, 1,
00146 Roma, ITALY, orlandi@mat.uniroma3.it , http://www.mat.uniroma3.it/users/orlandi/
http://arxiv.org/abs/0704.0582v1
http://www.math.rug.nl/~kuelske/
µε,Λ[η](dϕΛ)
〈i,j〉∈Λ V (ϕi−ϕj)−
i∈Λ,j∈Λc,|i−j|=1 V (ϕi)+
i∈Λ ηiϕi
i∈Λ(dϕi + εδ0(dϕi))
Zε,Λ[η]
where the partition function Zε,Λ[η] denotes the normalization constant that turns the
last expression into a probability measure. The Dirac-measures at the interface height
zero are multiplied with the parameter ε, having the meaning of a coupling strength.
The disorder configuration η = (ηi)i∈Rd denotes an arbitrary fixed configuration of
external fields, modelling a ”quenched” (or frozen) random environment.
What do we expect for such a model? Recall that the variance of a free massless
interface in a finite box diverges like the logarithm of the sidelength when there are no
random fields. Adding an arbitrarily small pinning ε (without disorder) always localizes
the interface uniformly in the volume, with the variance of the field behaving on the scale
| log ε| when ε tends to zero. Indeed, there is a beautiful and complete mathematical
understanding of the model without disorder, in the case of both Gaussian and uniformly
elliptic potentials (see [1, 7]) with precise asymptotics as the pinning force tends to zero.
These results follow from the analysis of the distribution of pinned sites and the random
walk (arising from the random walk representation of the covariance of the ϕi’s) with
killing at these sites. In this sense there is already a random system that needs to be
analyzed even without disorder in the original model.
What do we expect if we turn on randomness in the model and add the ηi’s ? Let
us review first what we know about the same model without a pinning force. In d = 2
we recently proved the deterministic lower bound µΛN [η](|ϕ0| ≥ t
logL) ≥ c exp(−ct2)
uniformly for any fixed disorder configuration η, for general potentials V (assuming not
too slow growth at infinity) [12]. So, it is not possible to stabilize an interface by cleverly
choosing a random field configuration (one could think e.g. that this might be possible
with a staggered field). As this result holds at any arbitrary fixed configuration here
we don’t need any assumptions on the distribution of random fields. This result clearly
excludes the existence of an infinite-volume Gibbs measure describing a two dimensional
interface in infinite volume in the presence of random fields. In another paper [8] the
question of existence of gradient Gibbs measures (Gibbs distributions of the increments
of the interface) in infinite volume was raised. Note that while interface states may not
exist in the infinite volume such gradient states may very well exist, as the example of
the two-dimensional Gaussian free field shows, by computation. (For existence beyond
the Gaussian case which is far less trivial, see [10, 11].) It was proved in [8] that there
are no such gradient Gibbs measures in the random model in dimension d = 2.
Now, turn to the full model in d = 2. In view of the localization taking place at any
positive pinning force ε without disorder, a natural guess might be that with disorder
at least at very large ε there would be pinning. However, we show as a result of the
present paper that this is not the case, somewhat to our own surprise, and an arbitrarily
strong pinning does not suffice to keep the interface bounded.
1.2 Main results
Delocalization in d = 2 - superextensivity of the overlap
Denote by ΛL the square of sidelength 2L+ 1 centered at the origin.
In this subsection we consider the disorder average of the overlap in ΛL showing
that it grows faster than the volume. This in particular implies that in two dimensions
there is never pinning, for arbitrarily weak random field and arbitrarily large pinning
forces ε. Here is the result.
Theorem 1.1 Assume that supt V
′′(t) ≤ 1, lim inf |t|↑∞
log V (t)
log |t|
> 1, and let ηi be sym-
metrically distributed, i.i.d. with finite second moment.
Let d = 2. Then there is a constant a > 0, independent of the distribution of the
random fields and the pinning strength ε ≥ 0, such that
lim inf
L2 logL
ηiµε,ΛL [η](ϕi)
≥ aE(η20). (2)
Note that the growth condition on V includes the quadratic case and ensures the
finiteness of the integrals appearing in (1) for all arbitrarily fixed choices of η, even at
ε = 0.
Generalizations to interactions that are non-nearest neighbor are obvious; all results
go through e.g. for finite range and we skip them in this presentation for the sake of
simplicity. We like to exhibit the case of Gaussian random fields (and not necessarily
Gaussian potential V ) since the bound acquires a form that looks even more striking
because it becomes independent of the size of the variance of the ηi’s (as long as this is
strictly positive).
Corollary 1.2 Let us assume that the random fields ηi have an i.i.d. Gaussian distri-
bution with mean zero and strictly positive variance of arbitrary size.
Then, with the same constant a as above, we have the bound
lim inf
L2 logL
µε,ΛL [η](ϕ
i )− µε,ΛL [η](ϕi)
≥ a > 0 (3)
for any 0 ≤ ε < ∞.
(3) follows from (2) by partial integration w.r.t. the Gaussian disorder average
(transforming the overlap into the variance of the ϕi’s).
Note that, even in the unpinned case of ε = 0, Theorem 1.1 is not entirely trivial in
the case of general potentials V . Here it provides an alternative simple way to see the
delocalization in the presence of random fields (while the explicit lower bound on the
tails of [12] provides more information.)
Lower bound on overlap in d ≥ 3
The analogue of Theorem 1.1 for higher dimensions is the following.
Theorem 1.3 Let d ≥ 3 and let ε ≥ 0 be arbitrary and assume the same conditions on
V and ηi as in Theorem 1.1.
Then there are positive constants B1, B2 < ∞, independent of the distribution of the
random fields and the pinning strength ε ≥ 0, such that
lim inf
ηiµε,ΛL[η](ϕi)
E(η20) (−∆−1)0,0
− log(B1 +B2ε) (4)
where the positive constant (−∆−1)0,0 is the diagonal element of the inverse of the
infinite-volume lattice Laplace operator whose existence is guaranteed in d ≥ 3.
Lower bound on the pinned volume in d ≥ 3
We complement the previous lower bounds on the overlaps which are depinning-type
of results by a pinning-type result. It is a lower bound on the disorder average of the
quenched Gibbs-expectation of the fraction of pinned sites. While we needed an upper
bound on the interaction potential V before we are assuming now a lower bound on V .
Theorem 1.4 Let d ≥ 3. Assume that inft V ′′(t) = c− > 0 and let ηi be symmetrically
distributed, i.i.d. with finite second moment.
Then there exist dimension-dependent constants C1, C2 > 0, independent of the
distribution of the disorder, such that, for all ε and for all volumes Λ, the disorder
average of the fraction of pinned sites obeys the estimate
µε,Λ[η](ϕi = 0)
≥ 1− C1 + C2E(η
log ε
. (5)
This shows pinning for the large ε regime in the ”thermodynamic sense” that the
fraction of pinned sites can be made arbitrarily close to one, uniformly in the volume.
As usual this result does not allow to make statement about the Gibbs measure itself.
The proofs follows from ”thermodynamic reasoning”. The first ”depinning-type”
result follows from taking the log of the partition function and differentiating and in-
tegrating back w.r.t. the coupling strength of the random fields. Exploiting the linear
form of the random fields, convexity, comparison of non-Gaussian with the Gaussian
partition functions, and asymptotics of Green’s functions the results follow, see Chapter
2 Proof of Depinning-type results
The estimates in formulas (2), (3), and (4) are immediate consequences of the following
fixed-disorder estimate.
Proposition 2.1 For any dimension d, there are constants CnG,d < ∞ and cG,d > 0
such that, for all fixed configurations of local fields η, we have
i,j∈Λ
(−∆Λ)−1i,j ηiηj − |Λ| log
CnG,d + ε
ηiµε,Λ[η](ϕi). (6)
Proof of the Proposition: Let us see what comes out when we differentiate and
integrate back the free energy in finite volume w.r.t. strength of the random fields.
logZε,Λ[hη] =
ηiµε,Λ[hη](ϕi). (7)
At every fixed η, this quantity is a monotone function of h, which is seen by another
differentiation w.r.t. h which produces the variance. We have
Zε,Λ[η]
Zε,Λ[0]
dhηiµε,Λ[hη](ϕi) ≤
ηi µε,Λ[η](ϕi). (8)
We note the lower bound on the numerator which we get by dropping the pinning term,
giving us
Zε,Λ[η] ≥ Zε=0,Λ[η]
≥ ZGaussε=0,Λ[η]
= exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
ZGaussε=0,Λ[0]
≥ exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
Here we have denoted by ZGaussε=0,Λ[η] the Gaussian partition function with potential V (t) =
Further we used that the lower bound on V (t) taken from the hypothesis implies
that, for any partition function in any volume D, we have Zε=0,D[0] ≤ C |D|nG,d. This gives
Zε,Λ[0] =
ε|A|Zε,Λ\A[0]
ε|A|C
|Λ\A|
nG,d = (CnG,d + ε)
So the desired estimate on the overlap follows from (8),(9),(10). This concludes the
proof of the Proposition. �
It is easy to obtain the Theorems 1.1 and 1.3 from the proposition. Indeed, taking
a disorder average we have
E(η20)
(−∆Λ)−1i,i − |Λ| log
CnG,d + ε
ηiµε,Λ[η](ϕi)
. (11)
Now use the asymptotics of the Green’s-function in a square (−∆ΛL)
i,i ∼ logL at fixed
i to get the first theorem. The proof of the case d ≥ 3 follows from the existence of the
infinite-volume Green’s-function in d ≥ 3.
Finally let us note in passing that a constant magnetic field is always winning against
an arbitrarily strong pinning, and even more strongly than a random field. Indeeed, let
d ≥ 2, let ηi = h ≥ 0 for all sites i and let ε ≥ 0 be arbitrary. Then, there is a constant
cd > 0, independent of h and ε, such that
lim inf
µε,ΛL [h](ϕi) ≥ cdh. (12)
This again follows from the Proposition, using
i,j∈Λ(−∆ΛL)
i,j ∼ Ld+2.
3 Proof of Pinning-type results
To prove the lower bound on the fraction of pinning sites in dimension d ≥ 3 given in
Theorem 1.4 we will in fact prove the following fixed-disorder lower bound:
For all finite volumes Λ and for all realizations η we have, for any ε0 > 0
µε,Λ[η](ϕi = 0)
log ε
2c−|Λ|
i,j∈Λ
(−∆Λ)−1i,j ηiηj
with a constant CG,d defined in (21).
Taking a disorder-expectation (5) follows by the finiteness of Green’s function in the
infinite volume (−∆
)−10,0 with ε0 = 1. �
Proof of (13): The proof is based on the trick to differentiate and integrate back
the log of the partition function, now w.r.t. ε: Differentiation gives
logZε,Λ[η] =
µε,Λ[η](ϕi = 0). (14)
We integrate this relation back, and it will be important for us to do it starting from a
positive ε0 > 0. So we get
Zε,Λ[η]
Zε0,Λ[η]
µε̃,Λ[η](ϕi = 0) ≤ log
µε,Λ[η](ϕi = 0) (15)
where we have used that
i∈Λ µε̃,Λ[η](ϕi = 0) is a monotone function of ε̃. Note that
the integrand itself is not a monotone function. (Compare [6] for a related non-random
pinning scenario, with back-integration from zero.)
Now we have the trivial lower bound obtained by keeping only the contribution in
the expansion where all sites are pinned, i.e.
Zε,Λ[η] ≥ ε|Λ|. (16)
For the upper bound on the partition function of the full model (at ε0) we first use
the lower bound on the potential V (t) ≥ c−t
giving us a comparison with a Gaussian
partition function with curvature c−:
Zε0,Λ[η] ≤ Z
Gauss,c−
[η]. (17)
It is a simple matter to rescale the Gaussian curvature away
Gauss,c−
[η] = c−
2 ZGauss
2 η] (18)
where the partition function on the r.h.s. is taken with unity curvature potential. For
the Gaussian partition function we claim the upper bound (writing again in the original
parameters) of the form
ZGaussε,Λ [η] ≤
ZGaussε=0,Λ[η]. (19)
Here is an elementary proof: We will replace successively the single-site integrations
involving the Dirac measure by integrations only over the Lebesgue measure with the
appropriately adjusted prefactor. Indeed, consider one site i and compute the contri-
bution to the partition function while fixing the values of ϕj for j not equal to i. Then
use that
dϕi + εδ0(dϕi)
ϕj + ηi)ϕi
= (2π)
2 exp
j∼i ϕj + ηi)
2 exp
j∼i ϕj + ηi)
dϕi exp
ϕj + ηi)ϕi
and iterate over the sites.
For the Gaussian unpinned partition function use
ZGaussε=0,Λ[η] = exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
ZGaussε=0,Λ[0]
≤ exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
with a suitable constant. From here (5) follows from (15,16,17,18,19,21) �
Acknowledgements: The authors thank Pietro Caputo for an interesting discus-
sion and Aernout van Enter for comments on a previous draft of the manuscript. C.K.
thanks the university Roma Tre for hospitality.
References
[1] E. Bolthausen, Y. Velenik, Critical behavior of the massless free field at the depinning
transition. Comm. Math. Phys. 223, 161-203, 2001.
[2] M. Biskup and R. Kotecký, Phase coexistence of gradient Gibbs states. Published Online
in Probab. Theory Rel. Fields DOI 10.1007/s00440-006-0013-6, 2007.
[3] A. Bovier and C. Külske, A rigorous renormalization group method for interfaces in random
media. Rev. Math. Phys. 6, 413–496, 1994.
[4] A. Bovier and C. Külske, There are no nice interfaces in (2 + 1)-dimensional SOS models
in random media, J. Statist. Phys., 83: 751–759, 1996.
[5] J. Bricmont, A. El Mellouki, and J. Fröhlich, Random surfaces in statistical mechanics:
roughening, rounding, wetting, . . . J. Statist. Phys. 42, 743–798, 1986.
[6] P. Caputo, Y. Velenik, A note on wetting transition for gradient fields. Stochastic Process.
Appl. 87, 107–113, 2000.
[7] J.-D. Deuschel, Y. Velenik, Non-Gaussian surface pinned by a weak potential. Probab.
Theory Related Fields 116, 359-377, 2000.
[8] A. C. D. van Enter, C. Külske, Non-existence of random gradient Gibbs measures in contin-
uous interface models in d = 2., math.PR/0611140, to be published in Annals of Applied
Probability
[9] G. Forgacs, R. Lipowski and Th.M. Nieuwenhuizen, The Behaviour of Interfaces in Ordered
and Disordered Systems, in Phase Transitions and Critical Phenomena, vol. 14, edited by
C. Domb and J.L. Lebowitz, Academic Press, 1986.
[10] T. Funaki, Stochastic Interface models. 2003 Saint Flour lectures, Springer Lecture Notes
in Mathematics, 1869, 103–294, 2005.
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face model. Comm. Math. Phys. 185, 1–36, 1997.
[12] C. Külske, E. Orlandi, A simple fluctuation lower bound for a disordered massless random
continuous spin model in d = 2. Electronic Comm. Probab. 11 200-205 (2006)
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112-169, 2006.
http://arxiv.org/abs/math/0611140
	Introduction
	The setup
	Main results
	Proof of Depinning-type results
	Proof of Pinning-type results

  We consider statistical mechanics models of continuous height effective
interfaces in the presence of a delta-pinning at height zero. There is a
detailed mathematical understanding of the depinning transition in 2 dimensions
without disorder. Then the variance of the interface height w.r.t. the Gibbs
measure stays bounded uniformly in the volume for any positive pinning force
and diverges like the logarithm of the pinning force when it tends to zero.
  How does the presence of a quenched disorder term in the Hamiltonian modify
this transition? We show that an arbitarily weak random field term is enough to
beat an arbitrarily strong delta-pinning in 2 dimensions and will cause
delocalization. The proof is based on a rigorous lower bound for the overlap
between local magnetizations and random fields in finite volume. In 2
dimensions it implies growth faster than the volume which is a contradiction to
localization. We also derive a simple complementary inequality which shows that
in higher dimensions the fraction of pinned sites converges to one when the
pinning force tends to infinity.

Introduction
1.1 The setup
The study of lattice effective interface models, continous and discrete, has a long tradi-
tion in statistical mechanics [14, 5, 9, 10, 13, 2, 3, 4].
The model we study is given in terms of variables ϕi ∈ R which, physically speaking,
are thought to represent height variables of a random surface at the sites i ∈ Zd.
Mathematically speaking they are just continuous unbounded (spin) variables. The
model is defined in terms of: a pair potential V , a quenched random term, and a
pinning term at interface height zero.
More precisely, we are interested in the behavior of the quenched finite-volume Gibbs
measures in a finite volume Λ⊂Zd with fixed boundary condition at height zero, given
University of Groningen, Department of Mathematics and Computing Sciences, Blauwborgje 3,
9747 AC Groningen, The Netherlands kuelske@math.rug.nl, http://www.math.rug.nl/∼kuelske/
Dipartimento di Matematica, Universit degli Studi ”Roma Tre”, Largo San Leonardo Murialdo, 1,
00146 Roma, ITALY, orlandi@mat.uniroma3.it , http://www.mat.uniroma3.it/users/orlandi/
http://arxiv.org/abs/0704.0582v1
http://www.math.rug.nl/~kuelske/
µε,Λ[η](dϕΛ)
〈i,j〉∈Λ V (ϕi−ϕj)−
i∈Λ,j∈Λc,|i−j|=1 V (ϕi)+
i∈Λ ηiϕi
i∈Λ(dϕi + εδ0(dϕi))
Zε,Λ[η]
where the partition function Zε,Λ[η] denotes the normalization constant that turns the
last expression into a probability measure. The Dirac-measures at the interface height
zero are multiplied with the parameter ε, having the meaning of a coupling strength.
The disorder configuration η = (ηi)i∈Rd denotes an arbitrary fixed configuration of
external fields, modelling a ”quenched” (or frozen) random environment.
What do we expect for such a model? Recall that the variance of a free massless
interface in a finite box diverges like the logarithm of the sidelength when there are no
random fields. Adding an arbitrarily small pinning ε (without disorder) always localizes
the interface uniformly in the volume, with the variance of the field behaving on the scale
| log ε| when ε tends to zero. Indeed, there is a beautiful and complete mathematical
understanding of the model without disorder, in the case of both Gaussian and uniformly
elliptic potentials (see [1, 7]) with precise asymptotics as the pinning force tends to zero.
These results follow from the analysis of the distribution of pinned sites and the random
walk (arising from the random walk representation of the covariance of the ϕi’s) with
killing at these sites. In this sense there is already a random system that needs to be
analyzed even without disorder in the original model.
What do we expect if we turn on randomness in the model and add the ηi’s ? Let
us review first what we know about the same model without a pinning force. In d = 2
we recently proved the deterministic lower bound µΛN [η](|ϕ0| ≥ t
logL) ≥ c exp(−ct2)
uniformly for any fixed disorder configuration η, for general potentials V (assuming not
too slow growth at infinity) [12]. So, it is not possible to stabilize an interface by cleverly
choosing a random field configuration (one could think e.g. that this might be possible
with a staggered field). As this result holds at any arbitrary fixed configuration here
we don’t need any assumptions on the distribution of random fields. This result clearly
excludes the existence of an infinite-volume Gibbs measure describing a two dimensional
interface in infinite volume in the presence of random fields. In another paper [8] the
question of existence of gradient Gibbs measures (Gibbs distributions of the increments
of the interface) in infinite volume was raised. Note that while interface states may not
exist in the infinite volume such gradient states may very well exist, as the example of
the two-dimensional Gaussian free field shows, by computation. (For existence beyond
the Gaussian case which is far less trivial, see [10, 11].) It was proved in [8] that there
are no such gradient Gibbs measures in the random model in dimension d = 2.
Now, turn to the full model in d = 2. In view of the localization taking place at any
positive pinning force ε without disorder, a natural guess might be that with disorder
at least at very large ε there would be pinning. However, we show as a result of the
present paper that this is not the case, somewhat to our own surprise, and an arbitrarily
strong pinning does not suffice to keep the interface bounded.
1.2 Main results
Delocalization in d = 2 - superextensivity of the overlap
Denote by ΛL the square of sidelength 2L+ 1 centered at the origin.
In this subsection we consider the disorder average of the overlap in ΛL showing
that it grows faster than the volume. This in particular implies that in two dimensions
there is never pinning, for arbitrarily weak random field and arbitrarily large pinning
forces ε. Here is the result.
Theorem 1.1 Assume that supt V
′′(t) ≤ 1, lim inf |t|↑∞
log V (t)
log |t|
> 1, and let ηi be sym-
metrically distributed, i.i.d. with finite second moment.
Let d = 2. Then there is a constant a > 0, independent of the distribution of the
random fields and the pinning strength ε ≥ 0, such that
lim inf
L2 logL
ηiµε,ΛL [η](ϕi)
≥ aE(η20). (2)
Note that the growth condition on V includes the quadratic case and ensures the
finiteness of the integrals appearing in (1) for all arbitrarily fixed choices of η, even at
ε = 0.
Generalizations to interactions that are non-nearest neighbor are obvious; all results
go through e.g. for finite range and we skip them in this presentation for the sake of
simplicity. We like to exhibit the case of Gaussian random fields (and not necessarily
Gaussian potential V ) since the bound acquires a form that looks even more striking
because it becomes independent of the size of the variance of the ηi’s (as long as this is
strictly positive).
Corollary 1.2 Let us assume that the random fields ηi have an i.i.d. Gaussian distri-
bution with mean zero and strictly positive variance of arbitrary size.
Then, with the same constant a as above, we have the bound
lim inf
L2 logL
µε,ΛL [η](ϕ
i )− µε,ΛL [η](ϕi)
≥ a > 0 (3)
for any 0 ≤ ε < ∞.
(3) follows from (2) by partial integration w.r.t. the Gaussian disorder average
(transforming the overlap into the variance of the ϕi’s).
Note that, even in the unpinned case of ε = 0, Theorem 1.1 is not entirely trivial in
the case of general potentials V . Here it provides an alternative simple way to see the
delocalization in the presence of random fields (while the explicit lower bound on the
tails of [12] provides more information.)
Lower bound on overlap in d ≥ 3
The analogue of Theorem 1.1 for higher dimensions is the following.
Theorem 1.3 Let d ≥ 3 and let ε ≥ 0 be arbitrary and assume the same conditions on
V and ηi as in Theorem 1.1.
Then there are positive constants B1, B2 < ∞, independent of the distribution of the
random fields and the pinning strength ε ≥ 0, such that
lim inf
ηiµε,ΛL[η](ϕi)
E(η20) (−∆−1)0,0
− log(B1 +B2ε) (4)
where the positive constant (−∆−1)0,0 is the diagonal element of the inverse of the
infinite-volume lattice Laplace operator whose existence is guaranteed in d ≥ 3.
Lower bound on the pinned volume in d ≥ 3
We complement the previous lower bounds on the overlaps which are depinning-type
of results by a pinning-type result. It is a lower bound on the disorder average of the
quenched Gibbs-expectation of the fraction of pinned sites. While we needed an upper
bound on the interaction potential V before we are assuming now a lower bound on V .
Theorem 1.4 Let d ≥ 3. Assume that inft V ′′(t) = c− > 0 and let ηi be symmetrically
distributed, i.i.d. with finite second moment.
Then there exist dimension-dependent constants C1, C2 > 0, independent of the
distribution of the disorder, such that, for all ε and for all volumes Λ, the disorder
average of the fraction of pinned sites obeys the estimate
µε,Λ[η](ϕi = 0)
≥ 1− C1 + C2E(η
log ε
. (5)
This shows pinning for the large ε regime in the ”thermodynamic sense” that the
fraction of pinned sites can be made arbitrarily close to one, uniformly in the volume.
As usual this result does not allow to make statement about the Gibbs measure itself.
The proofs follows from ”thermodynamic reasoning”. The first ”depinning-type”
result follows from taking the log of the partition function and differentiating and in-
tegrating back w.r.t. the coupling strength of the random fields. Exploiting the linear
form of the random fields, convexity, comparison of non-Gaussian with the Gaussian
partition functions, and asymptotics of Green’s functions the results follow, see Chapter
2 Proof of Depinning-type results
The estimates in formulas (2), (3), and (4) are immediate consequences of the following
fixed-disorder estimate.
Proposition 2.1 For any dimension d, there are constants CnG,d < ∞ and cG,d > 0
such that, for all fixed configurations of local fields η, we have
i,j∈Λ
(−∆Λ)−1i,j ηiηj − |Λ| log
CnG,d + ε
ηiµε,Λ[η](ϕi). (6)
Proof of the Proposition: Let us see what comes out when we differentiate and
integrate back the free energy in finite volume w.r.t. strength of the random fields.
logZε,Λ[hη] =
ηiµε,Λ[hη](ϕi). (7)
At every fixed η, this quantity is a monotone function of h, which is seen by another
differentiation w.r.t. h which produces the variance. We have
Zε,Λ[η]
Zε,Λ[0]
dhηiµε,Λ[hη](ϕi) ≤
ηi µε,Λ[η](ϕi). (8)
We note the lower bound on the numerator which we get by dropping the pinning term,
giving us
Zε,Λ[η] ≥ Zε=0,Λ[η]
≥ ZGaussε=0,Λ[η]
= exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
ZGaussε=0,Λ[0]
≥ exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
Here we have denoted by ZGaussε=0,Λ[η] the Gaussian partition function with potential V (t) =
Further we used that the lower bound on V (t) taken from the hypothesis implies
that, for any partition function in any volume D, we have Zε=0,D[0] ≤ C |D|nG,d. This gives
Zε,Λ[0] =
ε|A|Zε,Λ\A[0]
ε|A|C
|Λ\A|
nG,d = (CnG,d + ε)
So the desired estimate on the overlap follows from (8),(9),(10). This concludes the
proof of the Proposition. �
It is easy to obtain the Theorems 1.1 and 1.3 from the proposition. Indeed, taking
a disorder average we have
E(η20)
(−∆Λ)−1i,i − |Λ| log
CnG,d + ε
ηiµε,Λ[η](ϕi)
. (11)
Now use the asymptotics of the Green’s-function in a square (−∆ΛL)
i,i ∼ logL at fixed
i to get the first theorem. The proof of the case d ≥ 3 follows from the existence of the
infinite-volume Green’s-function in d ≥ 3.
Finally let us note in passing that a constant magnetic field is always winning against
an arbitrarily strong pinning, and even more strongly than a random field. Indeeed, let
d ≥ 2, let ηi = h ≥ 0 for all sites i and let ε ≥ 0 be arbitrary. Then, there is a constant
cd > 0, independent of h and ε, such that
lim inf
µε,ΛL [h](ϕi) ≥ cdh. (12)
This again follows from the Proposition, using
i,j∈Λ(−∆ΛL)
i,j ∼ Ld+2.
3 Proof of Pinning-type results
To prove the lower bound on the fraction of pinning sites in dimension d ≥ 3 given in
Theorem 1.4 we will in fact prove the following fixed-disorder lower bound:
For all finite volumes Λ and for all realizations η we have, for any ε0 > 0
µε,Λ[η](ϕi = 0)
log ε
2c−|Λ|
i,j∈Λ
(−∆Λ)−1i,j ηiηj
with a constant CG,d defined in (21).
Taking a disorder-expectation (5) follows by the finiteness of Green’s function in the
infinite volume (−∆
)−10,0 with ε0 = 1. �
Proof of (13): The proof is based on the trick to differentiate and integrate back
the log of the partition function, now w.r.t. ε: Differentiation gives
logZε,Λ[η] =
µε,Λ[η](ϕi = 0). (14)
We integrate this relation back, and it will be important for us to do it starting from a
positive ε0 > 0. So we get
Zε,Λ[η]
Zε0,Λ[η]
µε̃,Λ[η](ϕi = 0) ≤ log
µε,Λ[η](ϕi = 0) (15)
where we have used that
i∈Λ µε̃,Λ[η](ϕi = 0) is a monotone function of ε̃. Note that
the integrand itself is not a monotone function. (Compare [6] for a related non-random
pinning scenario, with back-integration from zero.)
Now we have the trivial lower bound obtained by keeping only the contribution in
the expansion where all sites are pinned, i.e.
Zε,Λ[η] ≥ ε|Λ|. (16)
For the upper bound on the partition function of the full model (at ε0) we first use
the lower bound on the potential V (t) ≥ c−t
giving us a comparison with a Gaussian
partition function with curvature c−:
Zε0,Λ[η] ≤ Z
Gauss,c−
[η]. (17)
It is a simple matter to rescale the Gaussian curvature away
Gauss,c−
[η] = c−
2 ZGauss
2 η] (18)
where the partition function on the r.h.s. is taken with unity curvature potential. For
the Gaussian partition function we claim the upper bound (writing again in the original
parameters) of the form
ZGaussε,Λ [η] ≤
ZGaussε=0,Λ[η]. (19)
Here is an elementary proof: We will replace successively the single-site integrations
involving the Dirac measure by integrations only over the Lebesgue measure with the
appropriately adjusted prefactor. Indeed, consider one site i and compute the contri-
bution to the partition function while fixing the values of ϕj for j not equal to i. Then
use that
dϕi + εδ0(dϕi)
ϕj + ηi)ϕi
= (2π)
2 exp
j∼i ϕj + ηi)
2 exp
j∼i ϕj + ηi)
dϕi exp
ϕj + ηi)ϕi
and iterate over the sites.
For the Gaussian unpinned partition function use
ZGaussε=0,Λ[η] = exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
ZGaussε=0,Λ[0]
≤ exp
i,j∈Λ
(−∆Λ)−1i,j ηiηj
with a suitable constant. From here (5) follows from (15,16,17,18,19,21) �
Acknowledgements: The authors thank Pietro Caputo for an interesting discus-
sion and Aernout van Enter for comments on a previous draft of the manuscript. C.K.
thanks the university Roma Tre for hospitality.
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http://arxiv.org/abs/math/0611140
	Introduction
	The setup
	Main results
	Proof of Depinning-type results
	Proof of Pinning-type results
