On an invariance principle for phase separation lines

On an invariance principle for phase separation lines

Lev Greenberg, Dmitry Ioffe

Faculty of Industrial Engineering, Technion, Haifa 3200, Israel Received 22 April 2004; accepted 22 July 2004

Available online 29 June 2005

Abstract

We prove invariance principles for phase separation lines in the two dimensional nearest neighbour Ising model up to the critical temperature and for connectivity lines in the general context of high temperature finite range ferromagnetic Ising models.

2005 Elsevier SAS. All rights reserved.

Résumé

Nous démontrons des principes d’invariance d’une part pour les lignes de séparation de phase, dans le cas du modèle d’Ising bidimensionnel à plus proches voisins jusqu’à la température critique, et d’autre part pour les lignes de connectivité dans le contexte général des modèles d’Ising ferromagnétiques à portée finie, pour les hautes températures.

2005 Elsevier SAS. All rights reserved.

MSC: 60F15; 60K15; 60K35; 82B20; 37C30

Keywords: Ising model; Random path representation; Ornstein–Zernike decay of correlations; Brownian bridge; Ruelle operator;

Renormalization; Local limit theorems

1. Introduction and results

1.1. Phase separation lines

A canonical example of phase separation lines is given in the framework of the nearest neighbour two dimen- sional Ising model below the critical temperatureT_c. It would be convenient to draw these lines through the bonds

✩ Partly supported by the EU Network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems under the contract HPRN-CT-2002-00277 and by the Senior Visiting Fellowship at the Isaac Newton Institute in the framework of the IGS Programme.

* Corresponding author.

E-mail address: ieioffe@ie.technion.ac.il (D. Ioffe).

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2005.05.001

of the integer latticeZ², which means that the corresponding spins should live on the vertices of the shifted dual latticeZ^2∗=(1/2,1/2)+Z². Let

S^∗_M=(1/2,1/2)+ [0, . . . , M−1] ×Z⊂Z^2∗

be an infinite dual strip of widthM. A spin configurationσ on S^∗_M is an element of{−1,1}^S∗M. Given a unit vector n∈S¹withn₁1/√

2 consider the so called Dobrushin’s boundary conditions

¯ σⁿ(y)=

+1, if(n^⊥, y)20,

−1, if(n^⊥, y)2<0 (1.1)

wheren^⊥=(−n₂,n₁)and(·,·)₂is the usual scalar product onR². In this way the configurationσ¯ⁿis defined on the whole ofZ^2∗ with all the spins above or on the line{y: (n^⊥, y)₂=0}being set to +1 and all the remaining spins being set to−1. Given a configurationσ∈ {−1,1}^S∗M consider a concatenation

σ_Mⁿ(y)=

σ (y), ify∈S^∗_M,

¯

σⁿ(y), ify∈Z^2∗\S^∗_M.

A bondb^∗=(x, y)ofZ^2∗is called frustrated inσ_Mⁿ ifσ_Mⁿ(x)=σ_Mⁿ(y). Each dual bondb^∗intersects (as an interval ofR²) a unique direct bondbofZ². Thus, everyσ∈ {−1,1}^S∗M gives rise to the set

BMⁿ(σ )= {b: b^∗is frustrated}.

Connected (as subsets ofR²) components ofBⁿ_M(σ )are called contours. These are microscopic boundaries between regions occupied with spins of different signs. Using a “rounding of corners” procedure [12] contours can be represented as either open or closed self-avoiding curves inR². Of course, all the contours ofσ are confined to the strip SM = [0, . . . , M] ×Z except for a unique infinite open contourγ which, outside SM is frozen by the Dobrushin’s boundary conditions (1.1) but, of course, varies inside S_M depending on a particular choice ofσ. This portion ofγ inside S_M which, with a slight abuse of notation, we shall continue to callγ is a self-avoiding line connecting the points 0 andu_N, where (Fig. 1)

M n₁n

= Nn=u_N. (1.2)

The contourγ models then-oriented microscopic interface between co-existing phases of the nearest neighbour Ising model at the inverse temperatureβ^∗> βc, whereβc=1/Tc is the phase transition threshold. The statistics Pⁿ_N,β∗ of γ is read from the Ising Gibbs distributionµⁿ_M,β∗ of σ∈ {−1,+1}^S∗M under Dobrushin boundary con- ditionsσ¯ⁿdefined in (1.1). The relation between the strip widthMand the projected length of the interfaceN is given by (1.2) above.

Fig. 1. Phase separation lineλ.

Our result, which we shall formulate in Subsection 1.6 asserts that underPⁿ_N,β∗ an appropriate rescaling of the interfaceγ converges to a Brownian bridge. We prove such convergence for any β^∗> β_c. The fact that the Brownian bridge picture persists all the way to the critical temperature was a well known conjecture.

Over the years statistical properties of phase separation lines have attracted a fair amount of attention. Apart from very specific exact computations of mean magnetization profiles (see e.g. [1,3]) and a study of a scaling limit for a simplified model of random strings [13], the Brownian bridge/random walk structure of interfaces in two dimensions has been previously investigated only for particular models in a perturbative (very low temperature) regime building upon the method of cluster expansions. Important results along these lines include a proof of Gaussian scaling for the interface height in [14] and a full invariance principle for the phase separation line in [15].

The local structure of very low temperature interfaces has been addressed in [5] and in [4].

Very low temperature cluster expansion based approach to a study of phase separation lines culminated in a series of works [12,9–11,16–18]. In these works fluctuations of phase separation lines have been investigated for Ising, Blume–Capel and lattice Widom–Rowlinson interfaces. The so called Dobrushin–Hryniv theory describes fluctuation of interfaces around limiting equilibrium crystal shapes and, furthermore, exactly quantifies the impact of the curvature of the equilibrium shape on the microscopic interface fluctuations in the corresponding directions.

In all the above works the role of very low temperature (very largeβ) was a purely technical one, for a broad class of models Dobrushin–Hryniv theory should be intrinsic for the whole of the (two) phase co-existing regime.

Indeed, in two dimensions interfaces look like essentially one dimensional aggregates of possibly complicated geometric objects and the fact that the original two-dimensional system is away from criticality should, in principle, find an expression in good mixing properties of the induced interface measures. In this respect interfaces in two dimensional low temperature models should resemble connectivity lines/clusters in high temperature models and, subsequently, a fluctuation analysis of the former should fit into a general framework of the Ornstein–Zernike theory.

Our results are based on a recent version of the Ornstein–Zernike theory developed in [6–8]. In the case of the two-dimensional nearest neighbour Ising model, due to very specific self-duality properties, there is a literal correspondence between phase separation lines in a low temperature model and connectivity lines of the high temperature dual model. In particular, an invariance principle for low temperature nearest neighbour Ising phase separation lines is equivalent to a modification of an invariance principle for high temperature connectivity lines in the context of nearest neighbour Ising models confined to lattice strips S_M.

In Subsection 1.5 we formulate an invariance principle for connectivity lines in the complete generality of high temperature finite range ferromagnetic Ising type models. A [6]-based proof of an invariance principle for subcritical percolation clusters has been previously given in [20].

Relevant facts from the Ornstein–Zernike theory [7] are collected in Section 2 and are used there for deriving sharp asymptotics for certain partition functions which are needed in order to conclude the proof in Section 3.

1.2. Random line representation onZ^d

We consider a class of finite range Ising models onZ^d,d2 with pair interactions given by the formal Hamil- tonian

(x,y)

J_|y−x|σ_xσ_y, (1.3)

where the coupling constants{Jx}are non-negative and the set{x: J_|x|=0}is bounded. To avoid trivialities we shall also assume that a random walk with jump rates{Jx}is irreducible. Letβcbe the inverse critical temperature corresponding to (1.3). For everyβ < βcthe following random line representation of two point functions is valid:

σ₀σ_xβ=

λ: 0→x

q_β(λ), (1.4)

where ·β it the unique infinite volume Gibbs state for the Hamiltonian (1.3) atβ. We refer to [7] for a discussion of (1.4), in particular for the compatibility conditions employed in the construction of pathsλ.

Givenn∈S^d−1, consider the setPⁿ_N of all compatible pathsλ: 0→u_N= Nn. The weightsq_β(·)induce a probability distribution onPⁿ_N:

Pⁿ_N,β(λ)= 1 σ₀σ_uNβ

q_β(λ). (1.5)

We would like to say that under the diffusive scaling, with the direction ofnplaying the role of time and the remaining(d−1)transverse dimensions playing the role of space, the family of measures{Pⁿ_N,β}weakly converges to the distribution of the(d−1)-dimensional Brownian bridge.

In order to make such a statement precise we need to recall some facts related to the geometry of the inverse correlation length and to the irreducible decomposition [7,8] of paths fromPⁿ_N.

1.3. The geometry of the problem

The inverse correlation lengthξ_β is defined via ξβ(y)= − lim

n→∞

1

nlog σ0σnyβ.

By [2],ξ_β is an equivalent norm onR^dfor everyβ < β_c. It is natural to study fluctuations of random linesλ∈Pⁿ_N in terms of the geometry ofξ_β. Define

Uβ

=

x∈R^d: ξ_β(x)1

and Kβ

=

n∈S^d−1

t∈R^d: (t, n)_dξ_β(n) .

U_β is, of course, just the unit ball in theξ_β norm, whereas K_β is the effective domain of the convex conjugate (Fenchel transform) ofξ_β. Clearly,ξ_β could be completely recovered from either of the two compact convex sets above. For example,

ξ_β(·)= max

t∈∂Kβ

(t,·)_d, (1.6)

where(·,·)_d stands for the usual scalar product onR^d. As it was proved in [7]∂K_β is locally analytic and has positive Gaussian curvature at anyt∈∂K_β for every finite range Ising model atβ < β_c. In particular, for every y∈R^d\0 the maximum in (1.6) is attained at the unique dual point t =t_y. For any t ∈∂K_β let κ_i =κ_i(t ), i=1, . . . , d−1, be the principal curvatures of∂K_β att. As we shall see below the magnitude of fluctuations of pathsλ∈Pⁿ_N underPⁿ_N,βdepends on the curvaturesκ₁, . . . , κ_d−1at the dual pointtn.

For the rest of this section let us a fix any particular model (interaction potential J= {J_y}) with the Hamil- tonian (1.3) and an inverse temperatureβ < βc(J).

1.4. Irreducible decomposition of paths

Letn∈S^d−1andt =t_n∈∂K_β is the dual point; (t,n)_d=ξ_β(n). Givenδ >0, a direction u∈R^d is called forward if

(t, u)_d> (1−δ)ξ_β(u).

LetCδ(t )be the cone of forward directions. In the sequel we shall fixδ small enough in order to ensure that any forward directionuhas a positive projection onn,(n, u)_d>0.

Given a compatible pathλ=(u₀, u₁, . . . , un)and a numberK, let us say thatul∈λis aK-correct break point if the following two conditions hold:

(A) (u_j,n)_d< (u_l,n)_d< (u_i,n)_dfor allj < l < i.

(B) The remaining sub-path(u_l+1, . . . , u_n)lies inside the set 2KU_β(u_l)+Cδ(t ).

A path is said to beK-irreducible if it does not containK-correct break point at all. We useSto denote the set of all irreducible paths (moduloZ^d-shifts). Define also the following three subsets ofS:

SL=

λ=(u₀, . . . , u_m)∈S: ∀l >0(u_l,n)_d< (u_m,n)_d , SR=

λ=(u₀, . . . , um)∈S: ∀l >0(ul,n)d> (u₀,n)dandλ⊂KUβ(u₀)+Cδ(t )

, (1.7)

S0=SL∩SR.

For any pathγ=(u₀, . . . , u_m)onZ^d setV (γ )=u_m−u₀. In other words,V (γ )is just the displacement alongγ. We employ the irreducible decomposition of pathsλ∈Pⁿ_N,

λ=η^Lγ₁ · · · γ_nη^R. (1.8)

For anyλ∈Pⁿ_N which has at least twoK-correct break points the decomposition (1.8) is unambiguously defined by the following set of conditions:

η^L∈SL, η^R∈SR and γ₁, . . . , γ_n∈S0.

In the above definition we have followed [8]. The only difference between (1.8) and the irreducible decomposition employed in [7] is that the break points here are defined with respect ton-orthogonal hyper-planes instead of tn-orthogonal hyper-planes. This is to ensure that the displacements along all theλ-paths which appear in (1.8) have positive projection on the direction ofn.

The renormalization calculus developed in [7] (see Theorem 2.3 there) implies that once the scaleKis chosen sufficiently large, pathsλ∈Pⁿ_Nwith only oneK-correct break point or withoutK-correct break points at all have exponentially smallPⁿ_N,β probabilities. From now on we shall tacitly assume that such large scaleKis fixed and, hence, the notion of irreducible decomposition (1.8) is well defined and the renormalization estimates of [7] apply.

1.5. Invariance principle for connection paths

Our approach to the invariance principle is based on the irreducible decomposition (1.8). The following two properties are crucial for the very formulation of the corresponding results:

min

V (η^L),n_d,

V (γ₁),n _d, . . .

V (γn),n _d,

V (η^R),n _d

>0 (1.9)

and

Pⁿ_N,β

diam(η^L)+max

i diam(γ_i)+diam(η^R) > (logN )² =O

1 N^ρ

, (1.10)

for anyρ >0.

Property (1.9) simply follows from the definition (1.7) of irreducible paths. Property (1.10) is a straightforward consequence of the abovementioned renormalization result (Theorem 2.3 in [7]) on the mass gap for irreducible connections.

DefineLN[λ]to be the linear interpolation inR^dthrough the vertices 0, V (η^L), V (η^L)+V (γ₁), . . . , V (η^L)+

n

1

V (γ_l),Nn.

By (1.9) the intersection number #(LN[λ] ∩Hⁿ_h)=1 for everyh∈ [0, (n,Nn)_d], where the(d−1)-dimensional hyper-planeHⁿ_h is defined via

Hⁿ_h=

u∈R^d: (u,n)_d=h .

Therefore, there is a natural parameterization ofLN[λ]as a functionΦ_Non the interval[0, (n,Nn)_d]: Φ_N(h)=LN[λ] ∩Hⁿh−hn.

Finally, we extend the domain ofΦ_N(·)to[0, N]by setting it to zero on((n,Nn)_d, N]. Notice thatΦ_N(·)takes values in the tangent spaceR^d−1to∂K_β at the dual pointt_n. From now on we shall record the values ofΦ_N(·)in the orthonormal coordinate system given by the principal curvature directionsv₁, . . . ,v_d−1of∂K_β att=t_n.

Since by (1.10) the Hausdorff distanced_H(·)betweenLN[λ]andλsatisfies, PⁿN,β

d_H

LN[λ], λ >

logN ² =O 1

N^ρ

,

any limit law forΦN(·)which arises on length scales much larger than(logN )²has a natural interpretation as a limit law for original pathsλunder the family of probability measures{Pⁿ_N,β}.

Define now the rescaled versionφN of the effective pathΦNas φ_N(h)= 1

√N Φ_N(N h); h∈ [0,1]. (1.11)

By construction,φ_N(h)∈C₀^d−1[0,1] = ×^d₀⁻¹C₀[0,1], where the latter is the space of continuous functions φ: 0→R^d−1satisfying the boundary conditionsφ(0)=φ(1)=0.

Theorem 1.1. The distribution ofφ_N(·)underPⁿ_N,β weakly converges onC₀^d−1[0,1]to the distribution of

√κ₁B₁(·), . . . ,√κ_d−1B_d−1(·) , (1.12)

whereB₁(·), . . . , B_d−1(·)are independent standard Brownian bridges on[0,1]andκ₁, . . . , κ_d−1are the principal curvatures of∂K_β att_n.

1.6. Phase boundaries in the nearest neighbour 2D Ising model

In the sequelβ^∗> βcis an inverse low temperature andβ < βcis the corresponding dual inverse (high) tempera- ture. We refer to [21,23] for a description of the duality relation between high and low temperature two dimensional nearest neighbour Ising models. In particular inverse correlation lengthξ_β(·)atβ equals to the surface tension τ_β∗(·)atβ^∗.

Let us proceed with the notation introduced in the opening Subsection 1.1. Recall that µⁿ_M,β∗ is the Gibbs measure on{−1,+1}^S∗M subject to Dobrushin’s boundary conditionsσ¯ⁿdefined in (1.1). The relation between the strip widthMand the projected interface lengthN was given in (1.2) and we usedPⁿ_N,β∗to denote the induced measure on crossing low temperature interfacesλ: 0→uN= Nn. However, the set of such interfaces is precisely the set of high temperature connectivity lines inside the direct strip SM. With a slight abuse of notation we shall continue to call this setPⁿ_N. Furthermore, by duality,

Pⁿ_N,β∗(λ)= 1 σ0σu_NN,β

q_N,β(λ). (1.13)

Apart from the fact that in the case of phase boundaries all our considerations are confined to two dimensions, the only difference with (1.5) is that now both the weightsqN,βand the expectation ·N,βcorrespond to the free Ising–

Gibbs state at the inverse temperatureβ on SM. As we shall see, however, the boundary effects have no impact on

the corresponding invariance principle. In order to decouple these effects from the predominant bulk behavior we shall need, though, a slightly different setup for the decomposition of paths (phase boundaries)λ∈Pⁿ_N. Namely, instead of (1.8) we shall employ:

λ=η^Lγη^R, (1.14)

where γ =(γ1, . . . , γn) is a string of elements from S0, whereas each of the left and right barriers η^L = (η^L₁, . . . , η_r^L)andη^R=(η^R₁, . . . , η_r^R)contain exactly

r=

(logN )²

elements withη^L₁ ∈SLand the rest belonging toS0.

As in the case of (1.10) the renormalization Theorem 2.3 of [7] and the strong triangle inequality of [19,23]

imply:

PⁿN,β^∗

diam(η^L)+diam(η^R)+max

i diam(γi) > (logN )⁴ =O

1 N^ρ

, (1.15)

for anyρ >0.

Similarly to the full-space case defineLN[λ]to be the linear interpolation inR^dthrough the vertices 0, V (η^L), V (η^L)+V (γ1), . . . , V (η^L)+

n

1

V (γl),Nn, and then define the rescaled effective pathφ_N(·)exactly as in (1.11).

By (1.15) the Hausdorff distanced_H(·)betweenLN[λ]andλis now bounded as, PⁿN,β

dH

LN[λ], λ > (logN )⁴ =O 1

N^ρ

,

which is still a vanishing quantity under the diffusive scaling.

Theorem 1.2. The distribution ofφ_N(·)underPⁿ_N,β∗weakly converges onC₀[0,1]to the distribution of 1

τ_β∗(x)+τ_β∗(x)

B(·) (1.16)

whereB(·)is the standard Brownian bridge andτ_β∗ is the surface tension of the dual low temperature model (considered here as a function on the unit circleS¹). Alternatively, (τ_β∗(x)+τ_β∗(x))⁻¹ is the curvature of the boundary∂K_βof the Wulff shape at the dual pointtn, ξ_β(n)=τ_β∗(n)=(tn,n)₂.

2. Decomposition of path weights

2.1. Properties ofq_β(·)-weights

We use notationqβ,Λ(·)for path weights which correspond to the Ising model with Hamiltonian (1.3) atβ < βc

with free boundary conditions on∂Λ. As before notationqβ(·)is reserved for infinite volume path weights . A basic reference for the definition and properties of these path weights and for the related path-compatibility conventions is [7,22,23]. As in [7] we shall repeatedly rely on the following decoupling property of theqβ(·)-weights (see Lemma 5.3 in [23] or the proof of Lemma 3.1 in [7]):

Lemma 2.1. Letγbe a (compatible) path andA, B⊆Z². Then, q_β,A(γ )

qβ,B(γ )exp

−c₁

x∈γ y∈AB

θ^|y−x|

, (2.17)

wherec₁=c₁(β)andθ=θ (β)∈(0,1).

Given two compatible pathsγ andλdefine the conditional weights q_β,A(γ|λ)=q_β,A(γ ∪λ)

qβ,A(λ) . By (2.17)

q_β,A(γ|λ) qβ,A(γ|η)exp

−c₁

x∈γ y∈λη

θ^|y−x|

. (2.18)

2.2. Strings of paths and semi-norm · θ

It would be convenient to consider countable stringsλof irreducible paths. Any finite stringλ=(λ₁, . . . , λ_n) is canonically extended to an infinite string by attaching from the right the dummy string of empty paths∅ = (∅,∅, . . .);

(λ₁, . . . , λ_n)→(λ₁, . . . , λ_n,∅,∅, . . .)=(λ,∅).

For any two countable stringsλ, αdefine the proximity index i( λ, α )=inf{k: λ_k=α_k},

and withθ=θ (β)∈(0,1)from (2.18) define the distance dθ( λ, α )=θ^{i( λ,α )−1}.

Let us say that a functionf is locally uniformly Lipschitz continuous if fθ

= sup

d_θ( λ,α )<1

|f ( λ )−f ( α )| dθ( λ, α ) <∞. 2.3. Uniform Ornstein–Zernike formula

Let us fix notation: unless it is explicitly mentioned, all paths or strings of paths are assumed to originate at 0.

Given a stringλand a pointu, let us useuλ(λu) to denote the shift ofλwhich starts (respectively ends up) inu.

Here is our main input from [7]: Given positive constantsa₁, a₂anda₃the OZ formula

γ:V (γ )=Nm

qβ

γ |V (γ )η f (γ )=C_f,η(m)

√N^d−1 e^{−N ξβ(m)}

1+o(1) , (2.19)

holds uniformly inm∈S^d−1∩Cδ(t ), boundary stringsηand functionsf satisfying 1

a₁f (·)a₂ and fθa₃. (2.20)

For every η and f the function C_f,η(·) is real analytic on S^d−1∩Cδ(t ) and, furthermore, there exist c₁ = c₁(a₁, a₂, a₃)andc₂=c₂(a₁, a₂, a₃)such that

0< c₁inf

f, ηC_f,η(·)sup

f, η

C_f,η(·)c₂<∞,

where both the inf and sup above are over all boundary stringsηand over all functionsf satisfying (2.20).

2.4. Decomposition ofqN(η^Lγη^R)

Letq_,β andq_,β be the path weights in the right (respectively left) half-spaces{z∈Z^d: z₁0}(respectively {z∈Z^d: z₁0}. Then, in view of the decoupling property (2.18), the super-logarithmic choice of the sizes of barriersη^Landη^Lin (1.14) ensures:

qN,β(η^Lγη^R) q_,β(η^L)q_,β(η^R) =q_β

γ|V (γ )η^R f_ηL(γ )

1+o(1), (2.21)

where

f_ηL(γ )=q_β(η^L|V (η^L)γ ) qβ(η^L) .

Notice thatf_ηLsatisfies (2.20) for all possible choices of the left barrierη^L. 2.5. Partition function Z_N(n)

By (1.10) we may restrict attention only to pathsλ: 0→u_N such that each irreducible piece in the decomposi- tion ofλhas a diameter bounded above by(logN )². In particular, we may assume that the diameters of the barriers η^Randη^Lin (2.21) are bounded above by(logN )⁴. By the uniform Ornstein–Zernike formula (2.19),

ZN(n)=

λ∈Pⁿ_N

q_N,β(λ)= C(n)

√N^d−1e^{−N ξβ(n)}

1+o(1), (2.22)

whereCis a strictly positive locally analytic function onS^d−1∩Cδ(t ),

C(n)=

η^L,η^R

q_,β(η^L)q_,β(η^R)exp

t, V (η^L)+V (η^R) C_f

ηL,η^R(n).

2.6. Decomposition ofq_β(γ^Lαγ^R|V (γ )η^R)

In order to prove convergence of finite dimensional distributions we shall consider a further splitting ofγ in (1.14) as

γ=γ^Lαγ^R,

where each of the three strings above contain at least(logN )²irreducible pieces. The induced decomposition of the conditional weights on the right-hand side of (2.21) is:

q_β(γ^Lαγ^R|V (γ )η^R)

q_β(α) =qβ

γ^L|V (γ

L)α γ^R|V (γ^R)η^R fα(γ^R). (2.23)

Again, notice thatf_α satisfies (2.20) for all possible choices of the barrierα. Furthermore, f_ηL(γ^Lαγ^R)=f_ηL(γ^L)

1+o(1), (2.24)

whenever the stringγ^Lcontains at least(logN )²irreducible pieces (which it will always do).

2.7. Partition function Z_N(n, v, α) Define the interior of the strip

S⁰_M=

y=(y1, y2)∈SM: min{y1, M−y1}> (logN )⁴ .

Let us fixv∈S⁰_M such that bothv andu_N−v belong to the forward coneCδ(t ). Fix also an irreducible string α=(α₀, α₁, . . . , α_{(logN )}2). The partition function Z_N(n, v, α)corresponds to the pathsλ: 0→u_N= Nnwhose irreducible decompositionη^Lγη^Rcontainsvαas a substring. By (2.21), (2.23) and (2.24),

Z_N(n, v, α )=g_α(n,n_v,n_uN−v)e^{−ξβ(v)−ξβ(uN−v)}q_β( α )e(t,V ( α ))

|v|^d−1|u_N−v|^d−1

1+o(1), (2.25)

where, as beforet=t_n,n_•= •/| • |and given two unit vectorsm,l∈Cδ(t )∩S^d−1, gα(n,m,l)=

η^L, η^R

q,β(η^L)q,β(η^R)exp

t, V (η^L)+V (η^R) Cf_ηL,α(m)C_fα,ηR(l).

3. Proof of the invariance principle

In this section we prove the invariance principle for phase boundaries as stated in Theorem 1.2. Since we rely on the high temperature dual representation (1.13) it would be convenient to writePⁿ_N,β=Pⁿ_N,β∗for the corresponding interface measures, where, of course,β^∗> β_c is an inverse low temperature whereasβ < β_c is the dual inverse high temperature. The general invariance principle for high temperature connectivity lines (Theorem 1.1) follows along the very same lines and, since we do not have to bother about boundary effects, is actually simpler. The corresponding multi-dimensional curvature computation could be found in [8].

3.1. The curvature computation

Letn=(n₁,n₂)∈S¹andn^⊥=(−n₂,n₁)be the unit orthogonal direction. Then, usingθ ()to denote the angle ofn+n^⊥;

θ ()=arctann₂+n₁ n₁−n₂

and recording the derivatives ofτ_β∗in it restriction to the unit circleS¹, we obtain:

τ_β∗(n+n^⊥)−τ_β∗(n)=

1+²τ_β∗

θ () −τ_β∗ θ (0)

=1 2²

τβ∗(n)+τ_β∗(n) +τ_β∗(n)

θ ()−θ (0) +o(²). (3.26)

Let nowu_N= Nnbe as in the statement of Theorem 1.2 and, given h∈(0,1)anda∈R, let v_N=N hn+

√N an^⊥ be an intermediate point inside the interior strip S⁰_M. Then (3.26) (notice thatθ ()+θ (−)−2θ (0)= o(²)) readily implies:

τ_β∗(v_N)+τ_β∗(u_N−v_N)−τ_β∗(u_N)

=hN

τ_β∗

n+ a h√

Nn^⊥

−τ_β∗(n)

+(1−h)N

τ_β∗

n− a (1−h)√

Nn^⊥

−τ_β∗(n)

=τ_β∗(n)+τ_β∗(n) 2h(1−h)

vN−N hn

√N

²+O 1

√N

. (3.27)

3.2. Finite dimensional distributions

We proceed with the setup of Subsection 1.6. Letκβ=(τβ∗(n)+τ_β∗(n))⁻¹be the curvature of∂Kβat the dual pointtn. In this section we shall prove convergence of one-dimensional distributions:

Lemma 3.1. For anyh∈(0,1)and anyf ∈C₀(R), lim

N→∞EⁿN,βf

φ_N(h) =Ef√κ_βB(h) . (3.28)

As it will be clear from the proof, a generalization to finite dimensional distributions of higher order is straight- forward.

It would suffice to show that one can find a sequence_N→0 such that for anyh∈(0,1), lim

N→∞

1 2_N

h+_N

h−_N

EⁿN,βf

φ_N(s) ds=Ef√κ_βB(h) . (3.29)

Givenh∈(0,1)anda∈Rdefine v_N=v_N(h, a)=N hn+√

N an^⊥. Thus,

φ_N(h)=a

=

v_N∈LN[λ]

=

λ∈Pⁿ_N(v_N) .

We record the decomposition (1.14) of each regular (that is satisfying the condition in (1.15)) pathλ∈Pⁿ_N(v_N)as, see Fig. 2,

λ=η^Lγ^Lαγ^Rη^R,

where the barrierα=(α₀, α₁, . . . , α_r)(recallr= (logN )²) is unambiguously defined by the condition vN∈

V (η^L)+V (γ^L), V (η^L)+V (γ^L)+V (α₀),

where[u, v)is the semi-open linear segment inR²with end-points atuandv. In the notation of Subsections 2.5 and 2.7,

Pⁿ_N,β

v_N∈LN[λ] =

(v,α ):

v_N∈[v,v+V (α₀))

ZN(n, v, α )

ZN(n) . (3.30)

By (1.15) we can restrict attention tov-s satisfying|v−v_N|(logN )⁴. Then the asymptotic expressions (2.22) and (2.25) and the curvature computation (3.27) yield:

Pⁿ_N,β

v_N∈LN[λ] 1+o(1) = 1

√N h(1−h)exp

− |vN−N hn|² 2N h(1−h)κ_β

(v,α ):

v_N∈[v,v+V (α₀))

G( α ), (3.31)

Fig. 2. Decomposition of a path λ∈Pⁿ_N(vN). The point vN belongs to the segment [V (η^L)+V (γ^L), V (η^L)+V (γ^L)+V (α0)).

α=(α₀, . . . , α_r)is a decoupling barrier.

whereG(α)=g_α(n,n,n)q_β(α)e(t,V ( α ))in the notation of (2.25).

Consequently, EⁿN,βf

φ_N(h) 1+o(1) =

(v,α ):

[v,v+V (α₀))∩HⁿN h=∅

f

v−(v,n)₂n

√N

e^{−|v−(v,n)2n|2/(2N h(1−h)κβ)}

√N h(1−h) G( α ),

where we have again relied on the possibility to restrict attention to|V (α0)|(logN )².

We are now in a position to derive (3.29): Choose_Nin such a way thatN _N(logN )². Then, 1+o(1)

h+N

h−_N

EⁿN,βf

φ_N(s) ds

= 1

√N h(1−h)

v,α

f

v−(v,n)₂n

√N

e^{−|v−(v,n)2n|2/(2N h(1−h)κβ)}G( α )

h+N

h−_N

1_{(v,n)₂sN(v+V (α₀),n)₂}ds

= C(n)

√h(1−h) 1 N^3/2

N (h−_N)(v,n)₂N (h+_N)

f

v−(v,n)₂n

√N

e^{−|v−(v,n)2n|2/(2N h(1−h)κβ)}+o(_N), where

C(n)=

α

V (α₀),n ₂G( α ). (3.32)

In fact,C(n)=1/√

2π+o(1). Indeed, the expression above is just the Riemannian sum for the integral 2_N

∞

−∞

f (z)e^{−z2/(2h(1−h)κβ)} 2h(1−h)κβ

dz+o(_N), and (3.29) follows.

3.3. Tightness

In this section we prove tightness of theϕ_N. In order to simplify the proof we introduce a modification ofϕ_N, which is a linear approximation ofϕ_N on a larger mesoscopic scale with step sizeS_N= N^1/2−δ.

The number of irreducible components in the decomposition (1.8) of a pathλbefore thei-th step on the meso- scopic grid will be denoted byT_N,i(λ)

T_N,i(λ)=max

k:

V (η^L),n ₂+ k

j=0

V (λ_j),n ₂iS_N

;

the coordinates of the displacement of the firstTN,i(λ)irreducible components in the pathλdenoted by C_N,i(λ)=V (η^L)+

T_N,i

j=0

V (λ_j).

Our mesoscopic linear approximation ofλ:0→u_N, which will be denoted by LN, is the linear interpolation through the points

0,V₁, . . . ,VN/S_N−1, uN∈R²,

where theV_i is recorded in(n,n^⊥)-coordinate system as:

Vi=iSNn+

CN,i−1(λ),n^⊥ ₂n^⊥.

The above definition of the vertices{V_i}is tuned in such a way that theirn(time) projections live on the step-S_N mesoscopic grid.

Similarly to the rescaled version of the effective pathϕ_N, we define a rescaled version of the approximation byϕ_N. Note that the irreducible decomposition procedure along with the restriction on the length of the irreducible components by(logN )⁴imply that

Nlim→∞dH(ϕN, ϕN)=0, (3.33)

where d_H(·,·)is Hausdorff distance.

To prove tightness ofϕ_N it is enough to show that

Eⁿ_N,βϕ_N(h₂)−ϕ_N(h₁)⁴C|h₂−h₁|² (3.34)

uniformly on 0h₁< h₂1.

Lemma 3.2. There existc₁, c₂>0 such that Pⁿ_N,βϕ_N(h)a√

h < c₁exp

−c₂min(a², a) (3.35)

holds uniformly onh∈(0,1)anda >0.

Proof. By definition ofϕ_N, we have Pⁿ_N,βϕ_N(h)a√

h =

(v,α ):

|(v,n^⊥)2|a√ hN (v,n)₂hN <(v+V (α₀),n)₂

ZN(n, v, α )

ZN(n) ; (3.36)

moreover it is sufficient to prove (3.35) forh∈(0,1)∩(Z·^S_N^N).