Full text of DOI:10.1016/S0304-4149(99)00113-1

Stochastic Processes and their Applications 87 (2000) 107–113
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A note on wetting transition for gradient ÿelds
P. Caputo ∗, Y. Velenik
Fachbereich Mathematik, Sekr. MA 7-4, TU-Berlin, Straꢀe des 17. Juni 136, D-10623 Berlin, Germany
Received 13 September 1999; received in revised form 2 November 1999; accepted 9 November 1999
Abstract
We prove existence of a wetting transition for two classes of gradient ÿelds which include: (1)
The Continuous SOS model in any dimension and (2) The massless Gaussian model in dimension 2.
Combined with a recent result proving the absence of such a transition for Gaussian models
above 2 dimensions (Bolthausen et al., 2000. J. Math. Phys. to appear), this shows in particu-
lar that absolute-value and quadratic interactions can give rise to completely diꢀerent behavior.
c
ꢀ 2000 Elsevier Science B.V. All rights reserved.
Keywords: Gradient models; Entropic repulsion; Pinning; Wetting transition
1. Introduction
In several recent papers (Bolthausen, 1999, Bolthausen et al., 2000, Bolthausen and
Brydges, 2000, Bolthausen and Ioꢀe, 1997, Deuschel and Velenik, 2000) the ques-
tion has been raised whether the two-dimensional massless Gaussian model exhibits a
wetting transition. It is well-known that the Gaussian ÿeld in 2D is delocalized, with
a logarithmically divergent variance, but that the introduction of an arbitrarily weak
self-potential favoring height 0 is enough to localize it, in the sense that the variance
remains ÿnite (Dunlop et al., 1992, Bolthausen and Brydges, 2000); this result has
recently been extended to a class of non-Gaussian models in a stronger form, showing
in particular existence of exponential moments for the heights (Deuschel and Velenik,
2000) and exponential decay of covariances (Ioꢀe and Velenik, 1998). In higher di-
mensions, the ÿeld is already localized without pinning potential, but the introduction
of such a potential turns the algebraic decay of the covariances into an exponential
one. On the other hand, a Gaussian ÿeld with a positivity constraint (“surface above
a hard-wall”) exhibits entropic repulsion. The average height diverges like log N in
p
2D (Deuschel and Giacomin, 1999) and
log N in higher dimensions (Bolthausen
et al., 1995) (N being the linear size of the box). When both a positivity constraint
and a pinning potential are present (“surface above an attractive hard wall”), there is
a competition between these two eꢀects. If there exists a (non-zero) critical value for
∗ Corresponding author. Fax: +49-30-31421695.
E-mail addresses: caputo@math.tu-berlin.de (P. Caputo), velenik@math.tu-berlin.de (Y. Velenik)
c
0304-4149/00/$ - see front matter ꢀ 2000 Elsevier Science B.V. All rights reserved.
PII: S0304-4149(99)00113-1

108
P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
the strength of the pinning potential above which the interface is localized, but below
which it is repelled by the wall, we say that the model exhibits a wetting transition.
That such a transition occurs in a wide class of 1D model is well-known, see e.g.
(Bolthausen et al., 2000, Burkhardt, 1981 and van Leeuwen and Hilhorst, 1981). It
was recently shown in Bolthausen et al. (2000) that the Gaussian model in dimensions
3 or higher does not display a wetting transition. The interface is always localized.
The physically important case of the 2D model (describing a 2D interface in a 3D
medium) remained however open.
In the present note, we prove that the 2D Gaussian ÿeld does exhibit a wetting
transition; in fact, the proof applies to any strictly convex interaction, see below. We
also prove that the continuous SOS model has such a transition in any dimension, thus
showing that the choice of the interaction can greatly aꢀect the physics of the sys-
tem. Our proofs are based on a variant of a beautiful and simple argument of Chalker
(1982), who proved the existence of a wetting transition in the discrete SOS model in
dimension 2.
2. Results
We consider a class of gradient models with single spin-space R+, i.e. modeling
surfaces above a hard wall. Let ꢀN ⊂ Zd be the cube of side N centered at the origin,
and
Hamiltonian:
HN0;a;b(ꢁ) = H00;N (ꢁ) + VNa;b(ꢁ);
: R → R an even function to be speciÿed later; we consider the following
where
X
X
x∈ꢀN ;y∈ꢀN
H00;N (ꢁ) =
(ꢁx − ꢁy) +
(ꢁx);
hx;yi
hx;yiX⊂ ꢀN
VNa;b(ꢁ) = −b
1
;
a; b ¿ 0;
{ꢁx6a}
x∈ꢀN
(hx; yi denotes a pair of nearest-neighbour sites). The corresponding Gibbs measure
(on (R+)ꢀ ) is then given by
N
−HN0; a; b(ꢁ)
ZN0;+;a;b
Y
e
ꢂN0;+;a;b(dꢁ) =
dꢁx:
x∈ꢀN
As in the pure pinning problem (i.e. without a wall) (Deuschel and Velenik, 2000),
the relevant parameter is ꢃ(a; b) = aeb and not both a and b separately. As usual, we
introduce the ꢄ-pinning limit, which is the model described by the measure
0
0; N (ꢁ)
e−H
Y
ꢂN0;+;ꢃ(dꢁ) = lim ꢂN0;+;a;b(dꢁ) =
(dꢁx + ꢃꢄ0(dꢁx)):
ZN0;+;ꢃ
a→0
x∈ꢀN
ꢃ(a;b)=ꢃ
Note that ꢂN0;+;ꢃ can be written more explicitly as
Z0;+
ZN0;+;ꢃ
X
ꢂN0;+;ꢃ(dꢁ) =
ꢃ|A| ꢀN \A ꢂꢀ0;+\A(dꢁ);
N
A ⊂ ꢀN

P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
109
where
and
Z
Y
Y
0
Zꢀ0;+
=
e−H
dꢁx
ꢄ0(dꢁy);
0; N (ꢁ)
N \A
y∈A
x∈ꢀN \A
Y
Y
0
ꢂꢀ0;+\A(dꢁ) = (Zꢀ0;+
)
−1 e−H
dꢁx
ꢄ0(dꢁy)
0; N (ꢁ)
N \A
N
y∈A
x∈ꢀN \A
is the Gibbs measure on ꢀN \A with zero boundary condition outside.
Remark. Here and everywhere else in this note, the integrals are restricted to the
positive real axis, so we do not write this condition explicitly.
A quantity of interest is the density of pinned sites, i.e. of those sites, where the
interface feels the eꢀect of the pinning potential; it is deÿned as
ꢅN (a; b) = |ꢀN |−1ꢂN0;+;a;b(ꢆN (ꢁ));
P
where ꢆN (ꢁ) =
1
. We also write ꢅ(a; b) = limN→∞ ꢅN (a; b). The cor-
{ꢁx6a}
x∈ꢀN
responding quantities in the ꢄ-pinning limit are denoted ꢅN (ꢃ), ꢅ(ꢃ) (measuring the
density of sites exactly at zero height). ꢅ will play the role of an order parameter for
the wetting transition.
Our results can then be stated as follows:
Theorem 2.1. Let (x) = |x|; d¿1. Suppose that aeb ¡ (2d)−1; then there exist two
constants C1(a; b; d) and C2(a; b; d) such that; for any M ¿ C1Nd−1
;
ꢂN0;+;a;b(ꢆN (ꢁ) ¿ M)6e−C M
:
2
In particular; ꢅ(a; b)=0. The corresponding results also hold in the case of ꢄ-pinning;
provided ꢃ ¡ (2d)−1
.
1
Theorem 2.2. Let (x) = x2; d = 2. There exists ꢃ0 ¿ 0 such that; for any ꢃ ¡ ꢃ0
2
there exist two constants C3 and C4 depending on ꢃ such that; for any M ¿ C3Nd−1
;
ꢂN0;+;ꢃ(ꢆN (ꢁ) ¿ M)6e−C M
:
4
In particular; ꢅ(ꢃ) = 0. The same also holds in the case of the square-well potential
provided aeb is small enough.
We recall that it is not hard to prove that ꢅ ¿ 0 when the pinning is strong. For
example, in the ꢄ-pinning case, we can proceed in the following way. Since,
Z
ZN0;+;ꢃ
ZN0;+;0
1
ꢃˆ
ꢃ
|ꢀN |−1log
=
ꢅN (ꢃˆ) dꢃˆ;
0
the result follows from ZN0;+;ꢃ¿ꢃ|ꢀ and the existence of a constant C such that
|
N
ZN0;+;06C|ꢀ |. To prove the latter inequality one can consider a shortest self-avoiding
N
path ! on Zd starting at some site of @ꢀN and containing all the sites of ꢀN , and use
P|!|−1
H00;N (ꢁ)¿
(ꢁ! − ꢁ!n+1 ).
n
n=1

110
P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
The above results imply the existence of a wetting transition in these models. 1
Together with the result of Bolthausen et al. (2000) that in the Gaussian model in
d¿3 there is no wetting transition, Theorem 2.1 shows a radical diꢀerence of behavior
between the Gaussian and the SOS interactions.
Remark. 1. It is not diꢁcult to see, looking at the proofs, that our theorems remain
true if we replace the SOS interaction (x) = |x| with any globally Lipschitz function
1
(not necessarily symmetric), and the Gaussian interaction (x) = x2 with any even,
2
convex
such that 1=c¿ 00(x)¿c for some c ¿ 0 and all x.
2. Even though the present work provides a proof of the wetting transition in the
models considered, several important issues remain completely open. In particular, it
would be most desirable to have a pathwise description of the ÿeld in both the localized
and repelled regimes, i.e. a proof that ꢅ ¿ 0 implies ÿniteness of the mean height of
any ÿxed spin in the thermodynamic limit (if possible with estimate on the tail and
exponential decay of correlations), and a proof that ꢅ = 0 implies that the mean height
of any ÿxed spin diverges (if possible with estimates on the rate). Another question
of physical interest would be to determine how the ÿeld delocalizes as the pinning
strength decreases to its critical value.
3. Proof of Theorem 2.1
We ÿrst consider the square-well potential. Let M ¿ 0; following (Chalker, 1982),
we introduce the set BM = {ꢁ: ꢆN (ꢁ)¿M}, and the set
(
)
X
CM
=
ꢁ:
1
¿M
:
{ꢁx62a}
x∈ꢀN
Since BM ⊂ CM , the ÿrst claim immediately follows from the estimate on conditional
probabilities
ꢂN0;+;a;b(BM |CM )6e−C M
;
M ¿ C1Nd−1
:
(3.1)
2
Moreover, ꢅ(a; b) = 0 will follow from this and the obvious bound
M
ꢅN 6
+ ꢂN0;+;a;b(BM );
Nd
by choosing M such that NdꢀMꢀNd−1
.
We turn to the proof of (3.1). We deÿne a map T from CM onto BM by
ꢀ
ꢁx
if ꢁx6a;
(Tꢁ)x =
ꢁx − a otherwise:
a; b
If we write e−V = 1
+ eb 1
and expand the corresponding products,
{ꢁx6a}
N
{ꢁx¿a}
we have
−HN0; a; b(ꢁ)
ZN0;+;a;b
Z
Y
e
ꢂN0;+;a;b(CM ) =
1
dꢁx
{ꢁ∈CM
}
x∈ꢀN
1 In the ꢄ-pinning case, it can easily be seen that ꢅ is monotonous in ꢃ, so that there is a single critical
value.

P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
111
Z
0
0; N (ꢁ)
e−H
X X
Y
=
eb|B|
1
dꢁx
{ꢁx6a}
ZN0;+;a;b
x∈B
A ⊂ ꢀN B ⊂ A
|A|¿M
Y
Y
×
1
dꢁy
1
dꢁz:
{2a¡ꢁz}
(3.2)
(3.3)
{a¡ꢁy62a}
y∈A\B
z∈ꢀN \A
Now, observe that
H00;N (ꢁ)6H00;N (T ꢁ) + da|@ꢀN | + 2da|B|
(@ꢀN being the set of x ∈ ꢀN neighbouring a site y ∈ ꢀN ). After the change of variables
˜
ꢁx = (Tꢁ)x, we have
ꢂN0;+;a;b(CM )
Z
0
˜
0; N (ꢁ)
e−H
X X
|
¿e−da|@ꢀ
(e−2daeb)|B|
A ⊂ ꢀN B ⊂ A
N
ZN0;+;a;b
|A|¿M
Y
Y
˜
˜
×
1{ꢁ 6a} dꢁx
1{a¡ꢁ } dꢁy
˜
˜
x
y
x∈A
y∈ꢀN \A
Z
0
˜
0; N (ꢁ)
e−H
X
|
|A|
= e−da|@ꢀ
eb|A|(e−2da + e−b
)
N
ZN0;+;a;b
A ⊂ ꢀN
|A|¿M
Y
Y
˜
˜
×
1{ꢁ 6a} dꢁx
1{a¡ꢁ } dꢁy
˜
˜
x
y
x∈A
y∈ꢀN \A
¿e−da|@ꢀ |(e−2da + e−b)M ꢂN0;+;a;b(BM );
N
where we used e−2da + e−b ¿ 1, which follows from aeb ¡ (2d)−1. This proves (3.1).
Let us now consider the case of the ꢄ-pinning potential. The proof is very similar.
Let M ¿ 0, and deÿne BM as in the previous case (but remember that now ꢆN is the
number of sites with height equal to 0). We also need a set analogous to the set CM
above: Let ꢂ ≡ (2d)−1 − ꢃ; we set
(
)
X
DM
=
ꢁ:
1
¿M
:
{ꢁx6ꢇ}
x∈ꢀN
We are going to show that
ꢂN0;+;ꢃ(BM |DM )6e−C
;
M ¿ C3Nd−1
:
(3.4)
M
4
As above, (3.4) is suꢁcient to prove our claims.
To prove (3.4) deÿne the map
ꢀ
ꢁx − ꢇ if ꢁx ¿ ꢇ
(Sꢁ)x =
0
otherwise

112
P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
from DM onto BM . Note that ꢂN0;+;ꢃ(DM ) can be written
Z
0
0; N (ꢁ)
e−H
X X
Y
ꢃ|B|
1
dꢁx
{ꢁx6ꢇ}
ZN0;+;ꢃ
A ⊂ ꢀN B ⊂ A
|A|¿M
x∈A\B
Y
Y
×
1
dꢁy
ꢄ0(dꢁz):
(3.5)
{ꢇ¡ꢁy}
z∈B
y∈ꢀN \A
We have
H00;N (ꢁ)6H00;N (Sꢁ) + ꢇd|@ꢀN | + ꢇ2d|A|;
˜
and therefore, letting ꢁx = (Sꢁ)x and integrating over the variables ꢁx, x ∈ A\B,
X X
ꢂN0;+;ꢃ(DM ) ¿ e−ꢇd|@ꢀ
e−ꢇ2d|A|ꢇ|A|−|B|ꢃ|B|
|
N
A ⊂ ꢀN B ⊂ A
|A|¿M
˜
Z
0
0; N (ꢁ)
e−H
Y
Y
˜
˜
×
dꢁx
ꢄ0(dꢁy)
ZN0;+;ꢃ
y∈A
x∈ꢀN \A
ꢁ
ꢂ
0;+
|A|
X
Z
ꢇ
ꢃ
ꢀN \A
|
|
= e−ꢇd|@ꢀ
ꢃ|A|e−ꢇ2d|A| 1 +
N
ZN0;+;ꢃ
A ⊂ ꢀN
|A|¿M
ꢁ
ꢁ
ꢂꢂM
ꢇ
ꢃ
¿ e−ꢇd|@ꢀ
e−ꢇ2d 1 +
ꢂN0;+;ꢃ(BM );
N
where we used e−ꢇ2d(1 + ꢇꢃ ) ¿ 1. This proves (3.4).
4. Proof of Theorem 2.2
We start with the ꢄ-pinning potential. We proceed as in the corresponding proof of
the previous section up to Eq. (3.5). To simplify notations, here we set ꢇ = 1. Writing
W = A ∪ ꢀcN , we have the estimate
X X
H00;N (ꢁ)6H00;N (Sꢁ) + 2|@ꢀN | + 8|A| + 2
(Sꢁ)y:
(4.6)
x∈@W y∈w
y∼x
P
P
Let us use the short-hand notation X(ꢁ) = 2
y∈W;y∼x ꢁy. Inserting this
x∈@W
˜
estimate in (3.5) and changing variables to ꢁx = (Sꢁ)x, we obtain
ꢁ
ꢁ
ꢂꢂ
0;+
|A|
X
Z
1
ꢃ
ꢂN0;+;ꢃ(DM )¿e−2|@ꢀ
ꢃ|A| e−8 1 +
ꢀN \A ꢂꢀ0;+\A(e−X(ꢁ)):
|
N
ZN0;+;ꢃ
N
A ⊂ ꢀN
|A|¿M
Since Jensen’s inequality implies that
−ꢂꢀ0; +\A(X(ꢁ))
ꢂꢀ0;+\A(e−X(ꢁ))¿e
;
N
N
the conclusion will follow as before, once we prove that
ꢂꢀ0;+ (X(ꢁ))6c1|@W|6c1(|@ꢀN | + |A|);
N \A

P. Caputo, Y. Velenik / Stochastic Processes and their Applications 87 (2000) 107–113
113
with c1, a ÿnite-independent constant. However, this follows immediately from the ex-
istence of the inÿnite-volume repulsed ÿeld pinned at the origin, which was established
in Dunlop et al. (1992) (Lemma 2:1). Notice that it is not the case in higher dimen-
sions, and therefore the argument does not apply. In fact, as was proved in Bolthausen
et al. (2000), there is no wetting transition in this case.
Let us ÿnally discuss the square-well case. Again we adapt the proof given in the
previous section. Consider expression (3.2). Now estimate (3.3) must be replaced by
the analogous of (4.6) for the square-well potential. In particular, we are led to prove
an upper bound for
ꢂꢀ0;+(X(ꢁ) | ꢁx6a; ∀x ∈ A; ꢁy ¿ a; ∀y ∈ A);
˜
N
P
P
c
˜
˜
where X(ꢁ) = 2a
y∈W;y∼x ꢁy, W = B ∪ ꢀN , for ÿxed B ⊂ A ⊂ ꢀN . However,
˜
˜
x∈@W
FKG inequality implies that this expectation increases if one raises both the boundary
conditions in ꢀcN and the conditioning in A to ꢁx = a, and modify the wall constraint
to ꢁy¿a for all y. Thus, after a last change of variables, we get
0;+
ꢂꢀ0;+(X(ꢁ) | ꢁx6a; ∀x ∈ A; ꢁy ¿ a; ∀y ∈ A)6ꢂ \A(X(ꢁ + a));
˜
˜
ꢀN
N
˜
and therefore – using |@W|6|@ꢀN | + |B| – conclusion (3.1) follows as above.
Acknowledgements
The authors would like to thank Dima Ioꢀe and Ofer Zeitouni for their comments
and interest in this work. They also thank the referee for useful remarks. Y.V. was
supported by DFG grant De 663=2-1.
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