NOTE ON BOLTHAUSEN-DEUSCHEL-ZEITOUNI’S PAPER ON THE
ABSENCE OF A WETTING TRANSITION FOR A PINNED HARMONIC CRYSTAL IN DIMENSIONS THREE AND LARGER
LOREN COQUILLE AND PIOTR MIŁO ´S
ABSTRACT. The article [1] provides a proof of the absence of a wetting transition for the discrete Gaussian free field conditioned to stay positive, and undergoing a weak delta-pinning at height 0.
The proof is generalized to the case of a square pinning-potential replacing the delta-pinning, but it relies on a lower bound on the probability for the field to stay above the support of the potential, the proof of which appears to be incorrect. We provide a modified proof of the absence of a wetting transition in the square-potential case, which does not require the aforementioned lower bound. This shows that the results of [1] hold true.
1. DEFINITIONS AND NOTATIONS
We keep the notations of [1] except for the field which we callφinstead ofX. LetAbe a finite subset ofZd, letφ=(φx)x∈Zd ∈RZ
d and the Hamiltonian defined as HA(φ)= 1
8d
X
x,y∈A∪∂A:|x−y|=1
(φx−φy)2 (1)
where∂Ais the outer boundary ofA. The following probability measure onRAdefines the discrete Gaussian free field onA(with zero boundary condition) :
PA(dφ)= 1
ZAe−HA(φ)dφAδ0(dφAc) (2) wheredφA=Q
x∈Adφxandδ0is the Dirac mass at 0. The partition functionZAis the normalization ZA = R
RAexp(−H(φA))dφA. We will also need the following definition of a setAbeing ∆-sparse (morally meaning that it has only one pinned point per cell of side-length∆), which we reproduce from [1, page 1215] :
Definition 1. LetN ∈ Z, ∆> 0,ΛN = {−bNc/2, . . . ,bNc/2}d and letl∆N = {zi}|li=∆N1| denote a finite collection of pointszi ∈ ΛN such that for eachy ∈ΛN∩∆Zdthere is exacly onez ∈l∆N such that
|z−y|<∆/10. LetAl∆
N = ΛN\l∆N.
2. LOWER BOUND ON THE PROBABILITY OF THE HARD WALL CONDITION
The proof of [1, Theorem 6] relies on [1, Proposition 3]. Unfortunately, the proof provided in the paper, when applied witht>0 provides a lower bound which is a little bit weaker than what is clamed, namely
Proposition 2. Correction of [1, Proposition 3] :
Assume d ≥ 3and let t ≥ 0. Then there exist three constants c1,c2,c3 > 0depending on t, and c4>0independent of t, such that, for all∆integer large enough
lim inf
N→∞ inf
l∆
N
1
(2N+1)d logPA
l∆ N
(Xi≥t,i∈Al∆
N)≥ −dlog∆
∆d +c1log log∆
∆d −c2ec4t
√
log∆
∆d(log∆)c3 (3) The statement of [1, Proposition 3] only contains the first two terms. The dependence in t vanishes between equations (2.4) and (2.5) in [1]. Note that fort = 0 the third term is irrelevant and the bound coincides with the one stated in the paper.
1
NOTE ON THE ABSENCE OF A WETTING TRANSITION FOR A PINNED GFF INd≥3 2
3. PROOF OF THE ABSENCE OF A WETTING TRANSITION IN THE SQUARE-POTENTIAL CASE
Let us introduce the following notations ξˆN = X
x∈ΛN
1[|φx|≤a], ξ˜N = X
x∈ΛN
1[φx∈[0,a]],
Ω+A ={φx ≥0,∀x∈A}, Ω+N ={φx ≥0,∀x∈ΛN} A={x∈ΛN :φx ∈[0,a]}
and the following probability meaure with square-potential pinning : P˜N,a,b(dφ)= 1
Z˜N,a,bexp
−H(φ)+ X
x∈ΛN
b1[φx∈[0,a]]
dφΛNδ0(dφΛc
N) in contrast with the measure used in [1] :
PˆN,a,b(dφ)= 1 ZˆN,a,bexp
−H(φ)+ X
x∈ΛN
b1[φx∈[−a,a]]
dφΛNδ0(dφΛc
N).
Observe that
P˜N,a,b( ˜ξN < Nd|Ω+N)= PˆN,a,b( ˆξN < Nd|Ω+N) Theorem 3. (Absence of wetting transition, [1, Theorem 6])
Assume d ≥3and let a,b>0be arbitrary. Then there existsb,a, ηb,a>0such that
P˜N,a,b( ˜ξN > b,aNd|Ω+N)≥1−exp(−ηb,aNd). (4) provided N is large enough.
Proof. Let us compute the probability of the complement event and provide bounds on the numer- ator and the denominator corresponding to the conditional probability :
P˜N,a,b( ˜ξN < Nd|Ω+N)= P˜N,a,b( ˜ξN < Nd∩Ω+N)
P˜N,a,b(Ω+N) (5)
3.1. Lower bound on the denominator.
P˜N,a,b(Ω+N)FKG≥ ZN
Z˜N,a,b X
A⊂ΛN
(eb−1)|A|PN(A ⊃A)
| {z }
(∗)
PN(Ω+Ac|A ⊃ A)
| {z }
≥PN(Ω+Ac|φ≡0onA)
=PAc(Ω+Ac)
PN(Ω+A|A ⊃A)
| {z }
=1
(6)
WritingA={x1, . . . ,x|A|},
(∗)= PN(φ∈[0,a]on A)=
|A|
Y
i=1
PN(φxi ∈[0,a]|φxi+1, . . . , φx|A| ∈[0,a]) (7)
≥
|A|
Y
i=1
PN(φxi ∈[0,a]|φxi+1, . . . , φx|A| =0) (8)
=
|A|
Y
i=1
P{xi+1,...,x|A|}c(φxi ∈[0,a])≥[c(1/2∧a)]|A| (9)
for somec=c(d)>0, since the variance of the free field is bounded ind ≥3.
Hence,
P˜N,a,b(Ω+N)≥ ZN
Z˜N,a,b X
A⊂ΛN
exp(J0|A|)PAc(Ω+Ac) (10) withJ0 =log(eb−1)+logc+log(1/2∧a).
NOTE ON THE ABSENCE OF A WETTING TRANSITION FOR A PINNED GFF INd≥3 3
3.2. Upper bound on the numerator.
P˜N,a,b( ˜ξN < Nd∩Ω+N)= ZN
Z˜N,a,b X
A:|A|<Nd
(eb−1)|A|P| N(A ⊃{z }A)
≤(1/2∧a)|A|
PN(Ω+N∩ {φ >a on Ac}|A ⊃A)
| {z }
≤1
(11)
≤ ZN
Z˜N,a,b]{A:|A|< Nd}exp(JNd) (12) withJ =log(eb−1)+log(1/2∧a), where]Xdenotes the cardinality of the setX.
3.3. Upper bound on(5).
P˜N,a,b( ˜ξN < Nd|Ω+N)≤ exp(JNd)]{A:|A|< Nd} P
A⊂ΛNexp(J0|A|)PAc(Ω+Ac) (13) And now we proceed similarily as for the proof withδ-pinning potential:
1
Nd log ˜PN,a,b( ˜ξN < Nd|Ω+N)= 1 Nd log
exp(JNd)]{A:|A|< Nd}
(14)
− 1
Nd log X
A⊂ΛN
exp(J0|A|)PAc(Ω+Ac) (15) The right hand side of (14) can be bounded by(J+1−log) asNtends to infinity (by a rough approximation and the Stirling formula), which in turn can be made as close to 0 as we want by choosing =(J) sufficiently small. See [1].
To bound (15) we use [1, Proposition 3] witht=0 which matches to our Proposition 2 : (15)≤ − 1
Nd log X
A⊂ΛN:A is∆−sparse
exp(J0|A|)PAc(Ω+Ac) (16)
≤ − 1 Nd
N
∆ d
[(dlog∆ +c0)+J0−dlog∆ +c1log log∆]
!
(17)
=−J0+c0+c1log log∆
∆d <0 for∆ = ∆(J0) large enough. (18) where ∆-sparseness corresponds to Definition 1 : a set A ⊂ ΛN is ∆-sparse if it equals Al∆
N, for some setl∆N.
REFERENCES
[1] E. Bolthausen, J. D. Deuschel, and O. Zeitouni. Absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger.J. Math. Phys., 41(3):1211–1223, 2000. Probabilistic techniques in equilibrium and nonequilibrium statistical physics.
L. COQUILLE, INSTITUTFOURIER, UMR 5582DUCNRS, UNIVERSITE DE´ GRENOBLEALPES, 100RUE DES
MATHEMATIQUES´ , 38610 GIERES` , FRANCE
E-mail address:loren.coquille@univ-grenoble-alpes.fr P. MIŁOS´, MIMUW, BANACHA2, 02-097 WARSZAWA, POLAND
E-mail address:pmilos@mimuw.edu.pl
