High points for the membrane model in the critical dimension∗

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.18(2013), no. 86, 1–17.

ISSN:1083-6489 DOI:10.1214/EJP.v18-2750

High points for the membrane model in the critical dimension

Alessandra Cipriani

Abstract

In this notice we study the fractal structure of the set of high points for the membrane model in the critical dimensiond= 4. The membrane model is a centered Gaussian field whose covariance is the inverse of the discrete bilaplacian operator onZ^{4. We} are able to compute the Hausdorff dimension of the set of points which are atypically high, and also that of clusters, showing that high points tend not to be evenly spread on the lattice. We will see that these results follow closely those obtained by O. Davi- aud [3] for the 2-dimensional discrete Gaussian Free Field.

Keywords: Membrane Model; extrema of Gaussian fields; bilaplacian; multiscale decomposi- tion; Hausdorff dimension.

AMS MSC 2010:60K35; 60G15; 60G60.

Submitted to EJP on April 17, 2013, final version accepted on September 21, 2013.

1 The model

The field of random interfaces has been widely studied in statistical mechanics.

These interfaces are described by a family of real-valued random variables indexed by the d-dimensional integer lattice, which are considered as a height configuration, namely they indicate the height of the interface above a reference hyperplane. The probability of a configuration depends on its energy (the Hamiltonian), which defines a measure on the space of such configurations. The most well-known models are the so-calledgradient model, in particular the Discrete Gaussian Free Field (DGFF), or har- monic crystal, whose Hamiltonian is a function of the discrete gradient of the heights, and themembrane model. The study of such interface was firstly undertaken by Saka- gawa in [8]; we are aware of the contributions of Kurt ([6], [7]) regarding also a phe- nomenon calledentropic repulsion in dimension4.

The Membrane Model is a Gaussian multivariate random variable whose Hamiltonian depends on the mean curvature of the interface, in particular favors configurations whose curvature is approximately constant. It is indeed a lattice-based scalar field {ϕx}_x∈Zd whereϕx∈Ris viewed as a height variable at the sitexof the lattice. There are three convenient and equivalent ways in which one can see such a field. Denote by V_N := [−N, N]^d∩Z^dthe centered box of side-length2N+ 1.Then

∗Support: Swiss National Science Foundation, grant 138141, and Forschungskredit (University of Zurich).

†Institut für Mathematik, Universität Zürich, Switzerland. E-mail:[email protected]

1. the membrane model is the random interface model whose distribution is given by

PN(dϕ) = 1 ZN

exp



−1 2

X

x∈Z^d

(∆ϕx)²



 Y

x∈VN

dϕx

Y

x∈∂2VN

δ0(dϕx), (1.1)

where∆is the discrete Laplacian,∂2VN :={y∈V_N^c : d(y, VN)≤2}andZN is the normalizing constant.

2. By re-summation, the lawPN of the field is the law of the centered Gaussian field onVN with covariance matrix

GN(x, y) :=CovN(ϕx, ϕy) = (∆²_N)⁻¹(x, y).

Here, ∆²_N(x, y) = ∆²(x, y)1_{x,y∈VN}is the Bilaplacian with 0-boundary conditions outsideV_N.

3. The model is a centered Gaussian field onVN whose covariance matrixGN satis- fies, forx∈VN,

∆²G_N(x, y) =δ_xy, y∈V_N G_N(x, y) = 0, y∈∂₂V_N.

Ford≥5the infinite volume Gibbs measureP exists [5, Prop. 1.2.3] and is the law of the centered Gaussian field with covariance matrix

G(x, y) = ∆⁻²(x, y).

The membrane model presents several points in common, as well as challenging differ- ences, from the more known DGFF. The former lacks some key features of the latter, namely

1. the random walk representation for the Green’s function. In the harmonic crystal, it is possible to establish the well-known relation involving the covariance matrix ΓN:

ΓN(x, y) =E^x





τ_∂VN−1

X

n=1

1_{Sn=y}



, (1.2)

whereE^xis the law of a standard random walk(Sn)_n≥0started atx∈Z^2andτ∂V_N

is the first exit time fromVN.

2. Absence of monotonicity, for example the FKG inequality.

It is thus not possible to rely on harmonic analysis to control the field, and this renders many problems solved for the harmonic crystal quite intractable. Despite the lack of such tools it is sufficient to establish two crucial properties to study the high points:

one is thelogarithmic boundon covariances which are explained in Lemma 2.1, and the other one is the2-Markov property, which can be stated as follows:

Definition 1.1(2-Markov property). LetA, B⊆VN anddist(A, B)≥3. Then{ϕx}_x∈A and{ϕx}_x∈B are independent under the conditional law

P_N(· |σ({ϕ_x, x /∈A∪B})).

This suggests that the behavior of certain Gaussian fields with respect to excee- dences is universal, in the sense that as soon as the model displays a Gibbs-Markov property and covariances decay at the same rate, then the behavior of high points is the same (with some small adjustments to be done according to the dimension). This

also opens up the question of whether there are other points in common between log- correlated Gaussian fields, and we believe a more precise answer will be given soon.

The starting point is understanding how many “high” points viz. points that grow more than the average there are typically. The first step is to find the average height of the field, in other words to show that there exists a constantc >0such that

E

x∈VmaxN

ϕx

/logN ^N→+∞−→ c.

Theorem 1.2([6, Theorem 1.2]). Letd= 4,`∈(0,1),

V_N<sup>`</sup> :={x∈V<sub>N</sub> : d(x, V<sub>N</sub><sup>c</sup>)≥`N} (1.3) and letg:= 8/π². Then

(a)

N→+∞lim P

sup

x∈VN

ϕx≥2p

2glogN

= 0.

(b) If0< ` <1/2,0< η <1there existsC=C(`, η)>0such that P sup

x∈V_N^`

ϕx≥ 2p

2g−η logN

!

≤exp −Clog²N .

Roughly said, the first-order approximation of the maximum is of orderlogN, which also implies that the field behaves approximately like independent variables. For us then anα-high point will be a point whose height is greater than 2√

2gαlogN. The behavior ofα-high points for the2-dimensional DGFF, as shown in [3], tells us that such points exhibit a fractal structure. Very similar results were obtained by Dembo, Peres, Rosen and Zeitouni in [4] for the set of late points of the 2-d standard random walk.

To begin with, we recall the definition of the discrete fractal dimension:

Definition 1.3 (Discrete fractal dimension, [1]). Let A ⊆ Z^d. If the following limit exists, the fractal dimension ofAis

dim(A) := lim

N→∞

log|A∩VN| logN . The fractal dimension of the high points is given then in Theorem 1.4(Number of high points). Let`∈(0,1), and

H_N(η) :=n

x∈V_N^` :ϕ_x≥2p

2gηlogNo be the set ofη-high points.

(a) For0< η <1we obtain the following limit in probability:

lim

N→+∞

log|HN(η)|

logN = 4(1−η²).

(b) For allδ >0there exists a constantC >0such that forN large PN

n|HN(η)| ≤N^{4(1−η2)−δ}o

≤exp(−Clog²N).

We can push further the comparison between the DGFF and the Membrane Model at their respective critical dimensions, and one can find an interesting similarity in the behavior of the points. [3] for example also showed that high points appear in clusters;

this is what occurs in the membrane model, as the following two theorems show:

Theorem 1.5(Cluster of high points 1). Let

D(x, ρ) :={y∈V_N : |y−x| ≤ρ}. For0< α < β <1andδ >0

N→+∞lim max

x∈V_N^` PN

HN(α)∩D(x, N^β)

logN −4β(1−(α/β)²)

> δ

= 0. (1.4) Theorem 1.6(Cluster of high points 2). For0< α <1,0< β <1andδ >0we have

lim

N→+∞max

x∈V_N^`

P

log|HN(α)∩D(x, N^β)|

logN −4β(1−α²)

> δ|x∈ H_N(α)

= 0.

It is also possible to evaluate the average number of pairs of high points as in the following theorem:

Theorem 1.7(Pairs of high points). Let0< α <1,0< β <1and let Fh,β(γ) :=γ²(1−β) +h(1−γ(1−β))²

β Γ_α,β:=

γ≥0 : 4−4β−4α²F_0,β(γ)≥0 =

γ≥0 : (1−α²γ²)≥0 , ρ(α, β) := 4 + 4β−4α² inf

γ∈Γ_α,βF2,β(γ)>0.

Note that Γ_α,β = [0, 1/α] is independent of β. Then the following limit in probability holds:

N→+∞lim log

(x, y)∈ HN(α) : |x−y| ≤N^β

logN =ρ(α, β).

Finally we can also show what the maximum width of a spike of given length is:

Theorem 1.8(The biggest high square). Let−1< η <1,DN(η)the side length of the biggest sub-box for which all height variables are uniformly greater than2√

2gηlogN, i. e.

DN(η) := sup

a∈N: ∃x∈V_N^` : min

y∈B(x,a)ϕy≥2p

2gηlogN

. Then the following limit in probability holds:

lim

N→+∞

logDN(η)

logN = 1−η 2 .

The paper is organized as follows: in Section 2 we will prove some preliminary results that will be used for the proofs of the main theorems, to which Section 3 is going to be devoted.

2 Preliminary Lemmas and results

Notation

D(x, a)(resp.D(x, a]) denotes the open (resp. closed) Euclidean ball of centerxand radiusa, whileB(x, a)is a box centered atxof side lengtha. For the rest of this notice, recall the definition (1.3) and we let once and for all`∈(0,1/2). Letx₀∈V_N and

M_α:=

x₀+i(N^α+ 4) : i∈N⁴andx₀+i(N^α+ 2)⊂V_N .

We denote byxB the center of a (sub)boxB and asΠα the union of sub-boxes of side- lengthN^α (without discretization issues) and midpoint inMα. Fα will be the sigma- algebra generated by{ϕx} forx∈S

B∈Π_α∂₂B. Practically we denote withΠ_α a set of disjoint boxes separated by layers of thickness 2, which thanks to the 2-Markov property will enable us to perform a decomposition procedure on these sets.

FurthermoreϕB:=E(ϕx_B|F∂₂B)andVarB(ϕx) :=VarN(ϕx|F∂₂B). 2.1 Lemmas

2.1.1 The functionG_N(·,·)

In order to prove some of the next results we will introduce the convolution of the har- monic Green’s function, which will prove to be a key tool to obtain the crucial estimates on the covariances of our model. LetAbe an arbitrary subset ofZ^{4, and for}x∈Alet ΓA(x,·)be the solution of the discrete boundary value problem

∆Γ_A(x, y) =δ_xy, y∈A Γ_A(x, y) = 0, y∈∂A.

Note thatΓN as in (1.2) is the unique solution to the above problem forA:=VN. The convolution ofΓN is

GN(x, y) := X

z∈V_N

ΓN(x, z)ΓN(z, y), x, y∈VN.

[6] contains several bounds and properties of such a function, and we would like here to recall those that we are going to use in the sequel: for allx, y∈VN

• symmetry:G_N(x, y) =G_N(y, x),

• [6, Lemma 2.2] if`∈(0,1/2)there existc<sub>1</sub>=c<sub>1</sub>(`)>0,c₂>0such that

glogN+c1≤GN(x, y)≤glogN+c2 (2.1) With this in mind it is now easier for us to show how to bound the variances and covari- ances of our field.

Lemma 2.1(Bounds on the variances). Letd= 4and0< δ <1. Then

• there existsC >0such that sup

x∈VN

Var_N(ϕ_x)≤glogN+C. (2.2)

• There existsC(`)>0such that sup

x∈V_N^`

|VarN(ϕ_x)−glogN| ≤C(`). (2.3)

• There existC >0andC(`)>0such that sup

x,y∈V_N^` x6=y

CovN(ϕx, ϕy)−g(logN−log|x−y|)≤C. (2.4)

sup

x,y∈V_N^` x6=y

|CovN(ϕx, ϕy)−g(logN−log|x−y|)| ≤C(`). (2.5)

Proof.For the variances see [6, Proposition 1.1]. For the covariances, remember that in [6, Corollary 2.9] that for alld≥4and for allx∈V_N^`

sup

y∈V_N^`

|GN(x, y)−GN(x, y)| ≤c=c(`)<+∞. (2.6) It is therefore sufficient to show that (2.4) and (2.5) hold forG(·,·). But we have from [6, Lemma 2.10], that there exists a constant K such that in d = 4 forx 6= y and all α∈(0,2)

G_N(x, x)−G_N(x, y) =glog|y−x|+K+o |y−x|^−α . Hence

GN(x, y)≤GN(x, y) +c=GN(x, x)−glog|y−x|+K⁰

(2.1)

≤

≤glogN−glog|y−x|+K⁰.

The other bound follows similarly by considering (2.5).

Next we give a decomposition of the field which is similar to the one existing for the DGFF (see for example [9, Section 2.1]). With this in mind, we can prove that conditioning on the values of the field assumed on the double boundary of a subset of VN ⊆Z⁴ (in fact of anyZ^d) the resulting field is again the membrane model restricted to the interior of the smaller domain.

Lemma 2.2. LetB⊆VN. LetF :=σ(ϕz, z∈VN\B). Then {ϕx}x∈B

=d {EN[ϕx|F] +ψx}_x∈B

where “=^d” indicates equality in distribution, in particular underPN(·) (a) ψx⊥⊥F^;

(b) {ψ_x}_x∈B is distributed as the membrane model with 0-boundary conditions on B.

Proof.Setψx := ϕx−E[ϕx|F]for allx∈ B. We have to show that the above results hold.

(a) It is clear from the definition.

(b) BeingPN a Gibbs measure, it satisfies the DLR equation: for allA⊆VN,FA^c :=

σ(ϕz, z∈A^c),

P_N(· |FA^c)(η) =P_A,η(·) P_N(dη)−a. s. (2.7) with

PA,η(dϕ) = 1 ZA

exp



−1 2

X

x∈Z^d

(∆ϕx)²



 Y

x∈A

dϕx

Y

x∈VN\A

δη_x(dϕx).

In other words, PA,η is a Gaussian distribution with covariance matrix ∆²_A−1

. SinceCovN(·,·|FA^c)we find out that it equals GA. In our case this means that CovN(·|F)is deterministic and equal toGB. So

CovN(ψx, ψy) =CovN(ψx, ψy|F) =CovN(ϕx, ϕy|F) =GB(x, y)

Remark 2.3. This result gives us a decomposition of the membrane model in all dimen- sions.

Lemma 2.4. Let0< α <1and0< β <1,δ >0and we define S =S() :=n

(x, y)∈V_N^` : N^β(1−)≤ |x−y| ≤N^βo .

Then there existC,₀>0(which can be chosen uniformly on(α, β)on compact sets of (0,1)⁴) andγ?:= 2(2−β)⁻¹such that for all≤0and allN

max

(x,y)∈SP(x, y∈ HN(α))≤CN^{−4α2F2,β(γ?)+δ}. Proof.LetZ:=ϕ_x+ϕ_y and we see that

{x, y∈ HN(α)} ⊆n

Z≥4p

2gαlogNo . We obtain also from (2.4) that

Cov_N(ϕ_x, ϕ_y)≤glogN−gβ(1−) logN+O(1).

Thus by (2.6) and (2.2)

Var_N(Z)≤(2g(2−β) +O() +O(1/logN)) logN.

SinceF_2,β(γ_?) =γ_?, using (2.8) P(Z ≥4p

2gαlogN)≤

≤ exp

− 16(√

2g)²α²log²N

2((2g(2−β) +O() +O(1/logN)) logN

≤

≤ exp −4α²γ^∗(1 +O() +O(1/logN)) logN

≤

≤ CN^{−4α2F2,β(γ?)+O()}.

Lemma 2.5. Let B := B(x,4N^β), > 0, b^±(α, β, , N) = 2√

2g(α(1−β)±) logN, I(α, β, , N) := [b⁻(α, β, , N), b⁺(α, β, , N)]. Then

max

x∈V_N^`

P(ϕB ∈/ I(α, β, , N))|ϕx≥2p

2gαlogN)^N→+∞−→ 0.

Proof.We shortenI,b⁺andb⁻for the above quantities. We recall here two useful facts about normal random variables (whose short proof is postponed to the appendix). If X ∼ N(0,1)then

P(|X| ≥a)≤exp(−a²/2), ∀a≥0, (2.8) P(|X| ≥a)≥exp(−a²/2)

√2πa , ∀a≥1. (2.9)

Forη >0we obtain with (2.8) and (2.9) P(ϕx≥2p

2gα(1 +η) logN|ϕx≥2p

2gαlogN)→0.

asN →+∞. This yields

P(ϕB∈/I|ϕx≥2p

2gαlogN) =o(1) + +P(ϕ_B ∈/I, ϕ_x≤2p

2gα(1 +η) logN|ϕ_x≥2p

2gαlogN)≤

≤o(1) +P(ϕB ∈/ I|ϕx∈(1,1 +η)2p

2gαlogN).

Now we write ϕ_x = ϕ_x −ϕ_B +ϕ_B and observe that ϕ_B ⊥⊥ ϕ_x−ϕ_B. Therefore Cov_N(ϕ_x, ϕ_B) =Var_N(ϕ_B)and so there existsZ∼ N(0, σ²_Z),σ_Z² >0, for which

ϕ_B= VarN(ϕB)

Var_N(ϕ_x)ϕ_x+Z, Z⊥⊥ϕ_x.

Ifxis the center ofB⊆Cwe can decompose the variances asVarC(ϕx) =VarC(ϕB) + VarB(ϕx), and with this

VarN(ϕB)

VarN(ϕx) = (1−β) +O 1

logN

. It must then be thatVarN(Z) =O(logN). Consequently

P(ϕB≥b⁺|ϕx∈(1,1 +η)2p

2gαlogN)≤

≤P

Z+

(1−β) +O 1

logN

(1 +η)2p

2gαlogN≥b⁺

→0 forη < /(α(1−β)). Similarly

P(ϕ_B ≤b⁻|ϕ_x∈(1,1 +η)2p

2gαlogN)≤

≤P

Z+

(1−β) +O 1

logN

2p

2gαlogN ≤b⁻

→0.

Lemma 2.6. We keep the notation of Lemma 2.4. Let 0 < α < β < 1 and δ > 0. For (x, y) ∈ S define T(x, y)as the set of sub-boxes of side length 2N^β such that the centered subbox of side lengthN^β containsx, y. Then we can findC, 0 >0 such that for≤₀and allN

max

x,y∈S B∈T(x,y)

P

{x, y∈ HN(α)} ∩n

ϕ_B ≤2p

2gαγ(1−β) logNo

≤CN^{−4α2F2,β(min{γ,γ?})+δ}.

₀can be chosen uniformly on(α, β)on compact sets of(0,1)⁴. Proof.Define

E:={x, y∈ HN(α)} ∩n

ϕ_B≤2p

2gαγ(1−β) logNo . We distinguish two cases:

γ≥γ?. We haveP(E)≤P({x, y∈ HN(α)}): the claim follows from Lemma 2.4 because min{γ, γ?}=γ?.

γ < γ?. It follows from the definition ofγ?that γ < γ? impliesγ <2(2−β)⁻¹. For this reason seta:= 1−γ(1−β)>0andb:=γ(2−β)−2<0. LettingZ :=a(ϕx+ϕy)+bϕB

E⊆n

Z ≥(2a+bγ(1−β))α2p

2glogNo . Furthermore we have the usual decomposition

VarN(Z) =a²VarN(ϕx) +a²VarN(ϕy) +b²VarN(ϕB) + +2abCov_N(ϕ_x, ϕ_B) + 2abCov_N(ϕ_y, ϕ_B) +

+2a²CovN(ϕx, ϕy). (2.10)

By Lemma 2.1

Var_N(ϕ_B) =Var_N(ϕ_xB)−Var(ϕ_xB|F∂2B)≤g(1−β) logN+O(1).

and

Cov_N(ϕ_x, ϕ_B) =E(E(ϕ_x|F∂2B)E(ϕ_xB|F∂2B)) =

=CovN(ϕx, ϕx_B)−Cov(ϕx, ϕx_B|F∂₂B)≥

≥g(logN−log|x−xB|)−g(βlogN−log|x−xB|) +O(1) =

=g(1−β) logN+O(1).

Analogously

CovN(ϕy, ϕB)≥g(1−β) logN+O(1).

Define the auxiliary function f(a, b, β) := 2a²(2−β) +b²(1−β) + 4ab(1−β). We use these bounds in (2.10) to obtain

Var_N(Z)≤(f(a, b, β) +O() +O(1/logN))glogN.

By the equality2a+b=γβ

4a²+b²+ 4ab= (2a+b)²=γ²β². Then

f(a, b, β) = (2a+b)²−β(2a²+b²+ 4ab) =

= (4a²+b²+ 4ab)(1−β) + 2βa²=

= (γβ)²(1−β) + 2βa²=

= β(βγ²(1−β) + 2a²) =

= β((2a+b)(1−a) + 2a²) =

= β(2a+b−ab).

Hence

VarN(Z)≤(β(2a+b−ab) +O() +O(1/logN))glogN. (2.11) Since2a+b−ab= 2a+bγ(1−β)(2.10) and (2.11) yield

P(E)≤Cexp

−

4α²(2a+b−ab)

β +O()

logN

. Finally notice that

βF2,β(γ) =βγ²(1−β) + 2(1−γ(1−β))²=βγ²(1−β) + 2a²=

= (2a+b)(1−a) + 2a²= 2a+b−ab.

This allows us to conclude the proof.

Finally we would like to recall

Lemma 2.7 ([6, Lemma 2.11]). Let 0 < n < N, A_N ⊆ Z⁴ be a box of side-length N, An ⊆AN a box of side-lengthn. Let0< < 1/2. There existsC >0 such that for all x∈Anwith|x−xB|< n

VarN(E(ϕx|F∂₂A_n)−EN(ϕx_B|F∂₂A_n)|F∂₂A_N)≤C.

Remark 2.8. In [6] the above Lemma is stated with the assumption that“the boxesAn

andAN have the same centerÂt’Ât’. However one sees that the result can be obtained removing this condition which is not necessary.

3 Five theorems

Proof of Theorem 1.4. The core of the proof is the lower bound (b) which was already proved by [6, Theorem 1.3] and is based on the hierarchical decomposition of the membrane model, similar to that of the DGFF (for the main idea supporting the proof we also refer to [2]). We show here for the reader’s convenience the upper bound, in order to obtain the desired limit in probability.

Proof of Theorem 1.4 (a).For anyδ >0one can apply Chebyshev’s inequality to get Pn

|HN(η)| ≤N^{−4(1−η2)−δ}o

≤N^{4(1−η2)+δ}E|HN(η)| ≤

≤ N^{−4(1−η2)−δ}N⁴max

x∈VN

P

ϕ_x≥2p

2gηlogN

≤

≤ N^{−4(1−η2)−δ}N⁴exp

− 8gη²log²N 2glogN+C

≤N^{−4(1−η2)−δ}N^4−4η2 →0 where we have used Lemma 2.1 too.

Proof of Theorem 1.5. We chooseη, δ >0and define D+:=n

ϕB ≤2p

2gηlogNo , C+:=n

|HN(α)∩D(x, N^β)| ≥N^{4β(1−(α/β)2)−δ)}o and for an >0to be fixed later

A:= [

y∈B(x,N^β)

n|E(ϕy|F∂₂B)−ϕB| ≥2p

2glogNo .

By Lemma 2.7VarN(ϕB−E(ϕy|F∂₂B))≤c(we may assume thatB(x, N^β)(V_N^`), and so

P(A) =O N^4βexp −clog²N

tends to 0. Furthemore alsoP(D^c₊)tends to0by virtue of the bounds on covariances and (2.8). We then have

P(C+) =E(P(C+|F∂₂B))≤P(A) +P(D^c₊) +E(P(C+|F∂₂B)1A^c∩D+)≤

≤o(1) +P

H_4Nβ

α−⁰ β

≥N^{4β(1−(α/β)2)−δ)}

where⁰satisfies

α−⁰

β log(4N^β) = (α−η−) logN.

By tuning the parametersN large enough andη, small enough we can obtain 4β 1−

α−⁰ β

²!

<4β 1− α

β ²!

+δ (roughly speaking, we have⁰≈α(1−β)). By Theorem 1.4

P

H_4Nβ

α−⁰ β

≥N^{4β(1−(α/β)2)−δ}

→0

and from this the claim follows. We now go to the lower bound proof, which is similar in spirit to the upper bound. By setting

D₋:=n

ϕ_B≥ −2p

2gηlogNo ,

C−:=n

|HN(α)∩D(x, N^β)| ≤N^{4β(1−(α/β)2)−δ)}o we also define

H^s_N(η) :=n

x∈V_N^s :ϕx≥2p

2gηlogNo

, s∈(0,1/2).

We observe that

P(C₋) =E(P(C₋|F∂2B))≤P(A) +P(D^c₋) +E(P(C₊|F∂2B)1_Ac∩D−)≤

≤o(1) +P

H^3/8_4Nβ

α+⁰ β

≤N^{4β(1−(α/β)2)−δ)}

where⁰satisfies

α+⁰

β log(4N^β) = (α+η+) logN and we conclude as before.

Proof of Theorem 1.6. We will use the notationb^±(α, β, η, N)as in the proof of Lemma 2.5. We will also introduce the following quantities: letB:=B(x,4N^β), and forη, δ >0,

E := n

|HN(α)∩D(x, N^β)| ≤N^{4β(1−α2)−δ}o ,

F :=

ϕB≥b⁻(α, β, η, N) , G := {x∈ H_N(α)}.

Lower bound. Thanks to the proof of Lemma 2.5 we haveP(E|G) =P(E|F∩G)P(F|G)+

o(1) =P(E|F∩G)(1 +o(1)) +o(1). This means that P(E|F, G) = P(E∩F∩G)

P(F∩G) ≤ 1 P(F∩G)

pP(G)P(E∩F) =

= 1

P(F|G)P(G)

pP(G)P(F)P(E|F) =

Lemma2.5

= (1 +o(1)) s

P(F)

P(G)P(E|F).

We know by the bounds (2.2) and (2.9)

P(G) = P(ϕx≥2p

2gαlogN)≥c1

exp

−_2g^8gα_log^2log_N+c^2N

2

c3logN ≥

≥ exp (−d⁰logN), P(F) = P(ϕB≥2p

2g(α(1−β)−η) logN)≤

≤ c4exp

−8g(α(1−β)−η)²log²N 2g(1−β) logN+c5

≤exp (−d⁰⁰logN)

for somed⁰, d⁰⁰>0. Therefore we can findd >0such thatP(F)/P(G)≤exp(dlogN) and to show the result it suffices to prove thatP(E|F)≤exp(−clog²N)for a posi- tivec. For this purpose define

A:= [

y∈B

n|E(ϕy|F∂₂B)−ϕB| ≥2p

2glogNo .

From Lemma 2.7 it follows thatP(A)≤exp(−clog²N)forc >0and from (2.9) that P(F)≥exp (−dlogN)for somed >0, all in allP(A|F)≤exp −O log²N

. So we

can write

P(E|F)≤P(F∩A) P(F) + +P(E∩F∩A^c)

P(F) ≤ exp −O log²N

+E(P(E|F∂2B)1_Ac1_F) P(F) . If we are onA^c∩F, then

P

|HN(α)∩D(x, N^β)| ≤N^{4β(1−α2)−δ}|F∂₂B

≤

≤P

H_4N^3/8β(α+⁰)

≤N^{4β(1−α2)−δ}

(3.1) where⁰ is such that

(α−(α(1−β)−η) +) logN = (α+⁰) log 4N^β. (3.2) From Theorem 1.4 we know that (3.1) is bounded from above byexp(−clog²N)for a constantc >0, provided that⁰ is small (which can be obtained ifη,andN are small, small and large respectively).

Upper bound. LetK∈N^and

β_j :=_K^jβ

1≤j≤K. Then let D1:=D x, N^β1

, Di:=D x, N^βi

\D x, N^βi−1 . SinceD x, N^β

=∪_1≤i≤NDi

n

HN(α)∩D x, N^β

≥N^{4β(1−α2)+}o

⊆

⊆ [

0≤i≤N

n|HN(α)∩D_i| ≥N^{4βi(1−α2)+/2}o

as soon asN is large. It is then sufficient to prove that for alli P

|H_N(α)∩D_i| ≥N^{4βi(1−α2)+/2}|x∈ H_N(α)_N→+∞

−→ 0.

We can considerβj’s for which4βj(1−α²) +/2≤4βj. LetBj :=B x,4N^βj , C:=n

|HN(α)∩Dj| ≥N^{4βj(1−α2)+/2}o andb⁺(α, βj, η, N)as above. By Lemma 2.5 we obtain

P C|x∈ HN(α)) =P(C∩

ϕB_j ≤b⁺(α, βj, η, N) |x∈ HN(α) +o(1).

If we setF :=

ϕ_Bj ≤b⁺(α, β_j, η, N) ,G:={x∈ HN(α)}we obtain P(C∩F|G)

Chebyshev inq.

≤ N^{−4βj(1−α2)−/2}

P(G) E(1F∩G|HN(α)∩Dj|) =

=N^{−4βj(1−α2)−/2} P(G) E



 X

y∈Dj

1_{{x,y∈HN(α)}}1F



≤

≤N^4βjα2−/2 P(G) sup

y∈Dj

P({x, y∈ HN(α)} ∩F). (3.3)

By the bounds on the covariance and the normal distribution we have

P(G)⁻¹≤N^4α2+/8 (3.4)

forN large. By Lemma 2.6 by definingγ^∗= _2−β²

j >1whenηis small andKlarge we obtain

sup

y∈Dj

P({x, y∈ HN(α)} ∩F)≤N^{−4α2F2,βj(1)+/8}=N^{−4α2(1+βj)+/8}. (3.5) Inserting (3.4) and (3.5) in (3.3) we obtain

P(C∩F|G)≤N^{4βjα2−/2+/8+4α2−4α2(1+βj)+/8}= 1 N^/4

N→+∞−→ 0

Proof of Theorem 1.7. Preliminary we would like to make some considerations. It holds thatρ(α, β)is positive and in particular

ρ(α, β)≥4 + 4β−4α²F2,β(1) = 4(1−α²)(1 +β). (3.6) (3.6) derives from the fact thatF2,β(γ)has a unique global minimum at1in the rangeγ∈ Γα, β. Moreover notice thatρ(α, β)is increasing inβ. If we setγm := inf_γ∈Γα,βF2,β(γ), γ∗:= infγ≥0andγ+ := sup Γα,β we haveγ∗ ≤γm≤γ+and moreover sinceFh,β(·)does not depend onαas well asΓ_α,β does not depend onβwe haveγ_m= min{γ∗, γ₊}. that

γ₊= 1/α≥1.

We are now ready to prove the lower and upper bounds.

Lower bound. We set C:=n

(x, y)∈ HN(α) : |x−y| ≤N^β ≤N^{ρ(α,β)−δ}o .

Setm_γ := 4−4β−4α²F_0,β(γ) = 4(1−β)(1−α²γ²)and chooseγ < γ₊(in order to havem_γ strictly positive). Further

F:=n

B ∈Π_β: ϕ_B≥2p

2gγ(1−β)αlogNo , D:=n

|F| ≥N^mγ−δ/2o . Theorem 1.4^(a) shows thatP(D^c)→0. Hence we rewrite

P(C) =o(1) +P(D∩C).

OnDwe have at least

B_j : 1≤j≤N^mγ−δ/2 boxes. Set D_j :=n

ϕ_Bj ≥2p

2gαγ(1−β) logNo . We observe

C∩D⊆E:=

N^mγ−δ/2

[

j=1

Dj∩n

|HN(α)∩Bj| ≤N^{(ρ(α,β)−mγ)/4−δ/8}o .

(a)The idea is to scale the square: now we take the box with meshN/N^βand the grid is made by{x_B:B∈ Πβ}. In this way Theorem 1.4 tells us thatH_N1−β(γα)≈N^{4(1−β)(1−γ2α2)}=N^mγ.

Let us now put for some arbitraryη >0 A:= [

B∈Πβ

[

y∈B(xB,N^β/2)

n|E(ϕy|FB)−ϕB| ≥2p

2gηlogNo .

As before P(A) =o(1) asN → +∞. Plugging this in, exactly as in the proof of Theorem 1.5

P(C∩D)≤o(1) +P(E∩A^c)≤

≤o(1) + +N^mγ−δ/2P

H^1/4_Nβ

α(1−γ(1−β)) +η β

≤N^{ρ(α,β)−mγ4 −δ8}

. Finally we observe that

ρ(α, β)−mγ

4 ≥2β 1−α²(1−γ(1−β))² β²

!

which isexp(−O(log²N))by Theorem 1.4 forηsmall enough, as we have already seen. HenceP(C∩D) =o(1), and we conclude the proof.

Upper bound. By Theorem 1.4 we see that forλ >0the number ofα-high points within distanceN^λβ is at mostN^{4(1−α2)+4λβ}. We have with (3.6) that4(1−α²) + 4λβ ≤ ρ(α, β)if

4(1−α²) + 4λβ≤4(1−α²)(1 +β) ⇐⇒ λ≤(1−α²).

Therefore when this condition is not satisfied it is enough to find that there exists h=h(δ)<1such that for allβ⁰∈[β(1−α²), β]

P n

(x, y)∈ HN(α) : N^β0 ≤ |x−y| ≤N^β0ho

≥N^{ρ(α,β0)+δ}

→0.

We separate the two casesγ_∗=γm: γ∗=γm. Define

E:=n

(x, y)∈ HN(α) : N^β0 ≤ |x−y| ≤N^β0h

≥N^{ρ(α,β0)+δ}o . By Chebyshev inequality

P(E)≤N^{−ρ(α,β0)−δ}E





X

(x,y):N^β0≤|x−y|≤N^β0h



1_{{x,y∈HN(α)}}≤

≤N^{−ρ(α,β0)−δ}N^{4+4β0−4α2F2,β0(γ∗)+δ/2},

where we have used the assumption thathis close to 1 and Lemma 2.4 γ_∗> γ_m. We construct for eachB∈Π_β⁰ a bigger box of size4N^β0 by juxtaposing

to it the 12adjacent subboxes of same side length. We call the set of such bigger boxesB, and for eachB⁰ ∈ B we center inxB⁰ a box of twice bigger volume asB⁰. The latter boxes belong to a new set namedC. We remark that all pairs of points within distanceN^β0 must belong to at least oneB⁰∈ B. For >0set

D:=

maxC∈CϕC≥(1 +α)(1−β⁰)2p

2glogN

.

By Lemma 2.1 and the fact that{ϕy: y∈B}with boundary conditions∂₂Bis a Gaussian field

P(D^c) ≤ |Π_β⁰|exp

−(1 +α)²(1−β⁰)²(2√

2g)²log²N 2glogN^β0+O(1)

→0 since|Πβ⁰|=O(N^4(1−β0)). So noticing thatα(γm+) = (1 +α)

P(E) =o(1) +P(E∩D)≤o(1) +N^{−ρ(α,β0)−δ}N^{4+4β0−4α2F2,β0(γm+)+δ/2} ifhis close to1. 4 + 4β⁰−4α²F2,β⁰(γm+)−→^→0ρ(α, β⁰), thusP(E)→0.

Proof of Theorem 1.8.

Lower bound. We recall the notation used in the proof of Theorem 1.4 by N. Kurt. For α∈(1/2,1)we choose1≤k≤K+ 1such that

αk := α(K−k+ 1)

K >1−η

2 −δ (3.7)

(δmust be thought small). Let us now define recursivelyΓα₁ := Πα₁. Then fori≥ 2, we setΓα_i as follows: for anyB ∈Γαi−1 defineΓB,α_i :={B⁰ ∈Πα_i: B⁰ ⊆B/2}. Then

Γ_αi:= [

B∈Γ_αi−1

Γ_B,αi.

We re-use the notationB^(k)for a sequence of boxesB1⊇B2⊇ · · · ⊇Bk,Bi∈Γα_i

for all1≤i≤k. Finally D_k:=n

B^(k): ϕ_Bi ≥(α−α_i)λ2p

2g(1−1/K) logN,∀1≤i≤Ko , C_k:={|Dk| ≥n_k}.

We denote the biggest box ofB^(k)withB_1,k. LetB be a box of side lengthN^αk/2 centered inB1,k. Letnk :=N^{κ+4α(k−1)(1−λ)2K} , whereκis the constant appearing in [7, Lemma 3.2]. Define moreover for >0

A:= [

y∈B

n|E(ϕy−ϕx_B|Fα_k)| ≥2p

2g(α−αk)(1−γK) logNo .

By Lemma 2.7P(A^c)→1andP(Ck)→1as in Theorem 1.4 (Ckis the same event).

So

P

DN(η)≤N^{1−η2 −δ}

≤

≤o(1) + +P

C_k∩A^c∩

miny∈Bϕ_y≥2p

2gηlogN

≤

Def. of A, D_k

≤ o(1) + +P

miny∈B(ϕy−E(ϕy|Fα_k))≤ 2p

2glogN(η−(α−α_k)(1−γ_K)(1−))

≤

≤P max

y∈V^1/4

Nαk

ϕy≥2p

2glogN(−η+ (α−αk)(1−γK)(1−))

!

where in the latter inequality we used the fact thatV_N^1/4_αk ⊇B. For 2p

2glogN(−η+ (α−αk)(1−γK)(1−))>2p

2glogN^αk (3.8) we would obtain thanks to Theorem 1.4 that forN large this probability tends to 0. But (3.7) and (3.8) give rise to a system of equations which has a solution for largeKandN,αclose to 1 andsmall when1/2 +η/2k/K < η/2 +δ+ 1/2. Upper bound. We setθ:= ^1−η₂ ,β :=θ+δ. We have first of all that

P



 [

B∈Πβ

{ϕB≥2p

2g(1−θ) logN}





N→+∞

−→ 0 (3.9)

since we have the variance bounds and (2.8). Furthermore let us define F :=







\

B∈Π_β

{ϕB ≤2p

2g(1−θ) logN}





 ,

C:=





 [

B∈Πβ

{∀x∈B(ϕx≥2p

2gηlogN)}





 . We then have

P(DN(η) ≥ N^θ+2δ

≤P(C)≤

≤P(F^c) +P(F∩C)≤

(3.9)

≤ o(1) +E(P(C|Fβ)1F).

If B ∈ Πβ we indicate with B^(1/4) the sub-box B(xB, N^β/2). Choose > 0 and define

A:= [

B∈ΠB

[

y∈B^(1/4)

n|E(ϕy−ϕx_B|F∂₂B)| ≥2p

2glogNo .

With Lemma 2.7 we obtain thatP(A)tends to0as in Theorem 1.5. We can further bound

P(D_N(η)≥N^θ+2δ)≤o(1) +E(P(C|Fβ)1_F∩Ac).

To go on we notice that P(C|Fβ)≤

N N^β

⁴

B∈Πmax_βP

∀x∈B(ϕ_x≥2p

2gηlogN)

(3.10) and in particular onF∩A^c

P

∀x∈B(ϕx≥2p

2gηlogN)

≤

≤P

∀x∈B(ϕx−E(ϕx|Fβ)≥2p

2glogN(η−(1−θ+)))|Fβ

=

=P max

x∈V^1/4

N β

ϕx≤2p

2glogN(θ+)

! . By Theorem 1.2 this quantity is O exp −dlog²N

for a positived when for in- stanceβ >(θ+)which implies < δ. To sum up

P(C|Fβ)≤exp 2(1−β) logN−dlog²N

→0 and recalling (3.10) we finish the proof.

A Gaussian bounds

Proof of(2.8)and(2.9).

(2.8) Fort > a >0,t+a > t−aand hencet²−a²>(t−a)², exp(a²/2)P(|X|> a) = 2 exp a²/2

P(X > a) =

= 2 Z +∞

a

√1 2πexp

−t²−a² 2

dt <

< 2 Z +∞

a

√1 2πexp

−(t−a)² 2

dt= 1.

Notice that the bound holds also at a=0.

(2.9) We have that the function

g(a) := 2P(X > a)−exp −a²/2

√ 2πa is such thatg(1)>0, and its derivative

g⁰(a) = 2

√

2πexp −a²/2

1 +a²−a³ a²

<0, ∀a≥1.

Sincelim_a→+∞g(a) = 0,g(a)is always non negative.

References

[1] M. Barlow and S. Taylor, Fractional dimension of sets in discrete spaces, J.Phys. A: Math.

Gen (1989), no. 22, 2621–2626. MR-1003752

[2] E. Bolthausen, J. D. Deuschel, and G. Giacomin,Entropic repulsion and the maximum of the two-dimensional harmonic crystal, The Annals of Probability29(2001), no. 4, 1670–1692.

MR-1880237

[3] O. Daviaud, Extremes of the discrete two-dimensional Gaussian Free Field, The Annals of Probability34(2006), no. 3, 962–986. MR-2243875

[4] A. Dembo, Y. Peres, J. Rosen, and O. Zeitouni,Late points for random walks in two dimen- sions, The Annals of Probability34(2006), no. 1, 219–263. MR-2206347

[5] N. Kurt,Entropic repulsion for a Gaussian membrane model in the critical and supercritical dimension, Ph.D. thesis, University of Zurich, 2008.

[6] , Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension, The Annals of Probability37(2009), no. 2, 687–725. MR-2510021

[7] N. Kurt, Entropic repulsion for a class of Gaussian interface models in high dimensions., Stochastic Processes Appl.117(2007), no. 1, 23–34. MR-2287101

[8] H. Sakagawa,Entropic repulsion for a gaussian lattice field with certain finite range inter- actions, J. Math. Phys.44(2003), no. 7, 2939–2951. MR-1982781

[9] A.-S. Sznitman, Topics in occupation times and Gaussian free fields., Zürich: European Mathematical Society (EMS), 2012 (English). MR-2932978

Acknowledgments. I would like to thank my supervisor Erwin Bolthausen for his guidance throughout this work. I am also grateful to the anonymous referee for his careful review of an earlier draft and for suggestions for improvement.

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