Abstract
We obtain sharp asymptotics for the probability that the \((2+1)\)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region \(\Lambda \), both under the infinite volume measure and under the measure with zero boundary conditions around \(\Lambda \), this probability turns out to behave like \(\exp (-\tau _\beta (0) L \log L )\), with \(\tau _\beta (0)\) the surface tension at zero tilt, also called step free energy, and L the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models (Bolthausen et al., Commun Math Phys 170(2):417–443, 1995; Deuschel et al., Stochastic Process Appl 89(2):333–354, 2000).
1 Introduction
Let \({\mathbb {P}}_\Lambda \) denote the Gibbs measure of the \((2+1)\)-dimensional SOS model on a box \(\Lambda \subset {\mathbb {Z}}^2\) with zero boundary condition. The configurations are discrete height functions \(\eta :\Lambda \mapsto {\mathbb {Z}}\) whereas \(\eta (x)=0\) for \(x\notin \Lambda \). The probability measure is given by
where \(\beta >0\) is the inverse temperature, and \(Z_\Lambda \) denotes the associated normalizing factor, called partition function. We will mostly consider the case where \(\Lambda =\Lambda _L=[-L,L]^2\cap {\mathbb {Z}}^2\) is the square of side \(2L+1\) in \({\mathbb {Z}}^2\) centered at the origin.
It is well known that, if \(\beta \) is sufficiently large (as we assume from here on), the limit of \({\mathbb {P}}_{\Lambda _L}\) as \(L\rightarrow \infty \) exists (in the sense that the probability of any local event converges), and is denoted \({\mathbb {P}}\), the infinite-volume Gibbs measure; see e.g. [3].
The infinite volume measure is characterized by the fact that heights have finite variance and exponentially decaying tails: the interface is globally very rigid and flat, the height is exactly zero on a set of sites of density \(1-O(\exp (-4\beta ))\) and typical fluctuations are isolated spikes; see [3, 4, 7]. The question we investigate here is that of large fluctuations of the interface, namely, the asymptotics of the probability that the interface is positive in a fixed large region. In order to formulate our main result, let us recall the definition of the surface tension at zero tilt, often referred to as step free energy:
Definition 1.1
Let \(\xi \) be the height function on \(\Lambda _L^c\) such that \(\xi (x)=1\) if \(x=(x_1,x_2)\) with \(x_2\geqslant 0\), and \(\xi (x)=0\) otherwise. Let \(Z_{\Lambda _L}^{\xi }\) be the partition function on \(\Lambda _L\) with boundary condition \(\xi \) (see Sect. 2.1 below for more details). Then, the surface tension at zero tilt is defined as
It is a known fact that \({\tau }_\beta (0)\) is well defined and that, for \(\beta \) sufficiently large, one has \({\tau }_\beta (0)>0\), see Lemma 2.4 below for more details. We have then:
Theorem 1.2
There exists \(\beta _0>0\) such that for any \(\beta \geqslant \beta _0\) one has
The same limit holds if we replace \({\mathbb {P}}_{\Lambda _L}\) by \({\mathbb {P}}\).
Actually, it will be clear from the proof that the result still holds if we replace the inequality \(\eta (x) \geqslant 0 \) with \(\eta (x)\geqslant n\), for any fixed \(n>0\).
We now describe the heuristics behind Theorem 1.2. In [7] (see also [6] for a summary of the main results) the scaling limit of the shape of the SOS surface in the box \(\Lambda _L\) with zero boundary conditions and conditioned to be non-negative was established in full detail. The SOS interface lifts rigidly to a height \(H(L)=\lfloor \tfrac{1}{4\beta }\log L\rfloor \), in order to create room for downward spike-like fluctuations (entropic repulsion). As a consequence there are H(L) macroscopic level lines, following approximately \(\partial \Lambda _L\), where the height of the surface jumps (roughly) by one. A fraction \(1-o(1)\) of the level lines is at distance o(L) from \(\partial \Lambda _L\) while the rest has a non trivial scaling limit as \(L\rightarrow \infty \), with flat and curved parts and 1 / 3 fluctuation exponent along the flat part. Roughly each of the level lines at distance o(L) from \(\partial \Lambda _L\) entails a surface energy cost \(|\partial \Lambda _L|\beta \tau _\beta (0)=8\beta L \tau _\beta (0)\). The total energy cost of the macroscopic level lines ensemble is therefore
which explains (1.2). The difficulty that arises in substantiating this heuristics is that the H(L) contours have mutual interactions. If these are naively estimated, they produce an additive term, of apriori indefinite sign, of order \(O(c_\beta |\partial \Lambda _L|H(L))=O(\varepsilon _\beta L\log L)\) in the energy cost. Here \(\varepsilon _\beta =c_\beta /\beta >0\) is a constant tending to zero as \(\beta \rightarrow \infty \), but non-zero for any finite \(\beta \). While this problem can be avoided when looking for a lower bound on the l.h.s. of (1.2), simply by imposing that the contours stay sufficiently far one from the other to neglect the interaction, as an upper bound we would get nothing better than \(-2\tau _\beta (0)+\varepsilon _\beta \).
The solution we find is an iterative monotonicity argument (Theorem 4.1), based on the FKG properties of the SOS model, which we believe is of interest by itself. This allows us to conclude that the possibly attractive effect of the mutual interaction potential is more than compensated by the loss of entropy due to the fact that the contours cannot mutually cross. As a consequence, the surface tension associated to n SOS contours is at least the sum of the individual surface tensions (Corollary 4.2).Footnote 1
1.1 Discussion
Since the early work of Lebowitz and Maes [14], the problem of computing the sharp large deviation behavior of the positivity event \(\eta (x)\geqslant 0\), \(x\in \Lambda _L\), has attracted much attention. Refined estimates have been obtained for continuous height models such as the Gaussian free field on \({\mathbb {Z}}^d\), see [1, 2, 8], as well as for more general lattice massless free fields [9]. A large deviation theory for such models was further developed in [10]. The problem is of particular relevance in the study of the entropic repulsion phenomenon [4], see e.g. [16] for a survey. Considerable progress has been recently made for the SOS model [5–7] and for the discrete Gaussian model [15] for which the SOS gradient term \(|\eta (x)-\eta (y)|\) in the energy function is replaced by \((\eta (x)-\eta (y))^2\), but the question of computing the limit in (1.2) remained unaddressed.
As a matter of comparison, let us briefly recall the known results for the two-dimensional continuous Gaussian case. If \({\mathbb {P}}_L\) denotes the 2D Gaussian free field on \(\Lambda _L\) with zero boundary condition, then for any \(\delta \in (0,1)\) one has
where \(\kappa (\delta )>0\) is a constant related to the relative capacity of the set \(\Lambda _{(1-\delta )L}\) with respect to \(\Lambda _L\) which satisfies \(\kappa (\delta )\rightarrow \infty \) as \(\delta \rightarrow 0\); see [1, Theorem 3]. On the other hand, boundary effects dominate if all heights in \(\Lambda _L\) are required to be nonnegative, and one expects [9, Section 3] that
for some \(\chi >0\). Because of its discrete nature, the SOS interface considered in our work presents a very different behavior. First, the rigidity of the interface allows one to consider the infinite volume limit—whereas the 2D massless free field does not admit such a limit. Second, while the typical height in the bulk under the positivity constraint is of order \(\log L\) just as in the case of the 2D massless free field, the cost of such a shift is much higher due to the unavoidable presence of as many as H(L) macroscopic level lines each of which has a definite cost proportional to the length. In particular, boundary terms do not dominate here and the estimate of Theorem 1.2 holds for \({\mathbb {P}}\) as well as for \({\mathbb {P}}_{\Lambda _L}\).
2 Contours, surface tension, etc.
Here we define the model, and the notion of contours of the SOS interface. To express the law of contours we shall use a cluster expansion for partition functions of the SOS model. Finally we recall the definition of surface tension for a general tilt, and some of its properties.
2.1 SOS model: basic definitions and notation
We call a bond (resp. dual bond) any straight line segment joining two neighboring sites in \({{\mathbb {Z}}^2}\) (resp. of \({{\mathbb {Z}}^2}^*\), the dual lattice of \({\mathbb {Z}}^2\)). Here \({\mathbb {Z}}^2\) and \({{\mathbb {Z}}^2}^*\equiv {\mathbb {Z}}^2+(1/2,1/2)\) are thought of as embedded in \({\mathbb {R}}^2\). For any finite \(\Lambda \subset {\mathbb {Z}}^2\), let \({\mathcal {B}}_\Lambda \subset {{\mathbb {Z}}^2}\) denote the set of bonds of the form \(e=xy\) with \(x\in \Lambda \) and \(y \in \Lambda \cup \partial \Lambda \), where \(\partial \Lambda \) is the external boundary of \(\Lambda \), i.e. the set of \(y\in \Lambda ^c\) such that xy is a bond for some \(x\in \Lambda \). A height configuration \({\tau }: \Lambda ^c\mapsto {\mathbb {Z}}\) is called a boundary condition. We define \(\Omega _\Lambda ^{\tau }\) as the set of height functions \(\eta :{\mathbb {Z}}^2\mapsto {\mathbb {Z}}\) such that \(\eta (x)={\tau }(x)\) for all \(x\notin \Lambda \). The SOS Hamiltonian in \(\Lambda \) with boundary condition \({\tau }\) is the function defined by
The SOS Gibbs measure in \(\Lambda \) with boundary condition \({\tau }\) at inverse temperature \(\beta \) is the probability measure \({\mathbb {P}}_\Lambda ^{\tau }\) on \(\Omega _\Lambda ^{\tau }\) given by
where \(Z_\Lambda ^{\tau }\) is the partition function
When \({\tau }=0\) we simply write \(Z_\Lambda \) for \(Z_\Lambda ^0\) and \({\mathbb {P}}_\Lambda \) for \({\mathbb {P}}_\Lambda ^0\). We often consider boxes \(\Lambda \) of rectangular shape, and write \(\Lambda _{L,M}\), with \(L,M\in {\mathbb {N}}\), for the rectangle \(\Lambda _{L,M}=([-L,L]\times [-M,M])\cap {\mathbb {Z}}^2\) centered at the origin. When \(L=M\) we write \(\Lambda _L\) for the square of side \(2L+1\).
We recall that the SOS model satisfies the so called FKG inequality [12] with respect to the natural partial order defined by \(\eta \leqslant \eta '\Leftrightarrow \eta (x)\leqslant \eta '(x)\) for every x. That is, if f and g are two increasing (w.r.t. the above partial order) functions, then \({\mathbb {E}}^{\tau }_\Lambda (fg)\ge {\mathbb {E}}^{\tau }_\Lambda (f) {\mathbb {E}}^{\tau }_\Lambda (g)\) for any region \(\Lambda \) and any boundary condition \({\tau }\), where \({\mathbb {E}}^{\tau }_\Lambda \) denotes expectation w.r.t. \({\mathbb {P}}_\Lambda ^{\tau }\). To prove the FKG inequality one can establish directly the validity of the FKG lattice condition
2.2 Geometric contours, h-contours etc.
We use the following notion of contours.
Definition 2.1
Two sites x, y in \({\mathbb {Z}}^2\) are said to be separated by a dual bond e if their distance (in \({\mathbb {R}}^2\)) from e is \(\tfrac{1}{2}\). A pair of orthogonal dual bonds which meet in a site \(x^*\in {{\mathbb {Z}}^2}^*\) is said to be a linked pair of bonds if both are on the same side of the forty-five degrees line (w.r.t. to the horizontal axis) across \(x^*\). A geometric contour (for short a contour in the sequel) is a sequence \(e_0,\ldots ,e_n\) of dual bonds such that:
-
(1)
\(e_i\ne e_j\) for \(i\ne j\), except for \(i=0\) and \(j=n\) where \(e_0=e_n\).
-
(2)
for every i, \(e_i\) and \(e_{i+1}\) have a common vertex in \({{\mathbb {Z}}^2}^*\).
-
(3)
if \(e_i,e_{i+1},e_j,e_{j+1}\) all have a common vertex \(x^*\in {{\mathbb {Z}}^2}^*\), then \(e_i,e_{i+1}\) and \(e_j,e_{j+1}\) are linked pairs of bonds.
We denote the length of a contour \(\gamma \), i.e. the number of distinct bonds in \(\gamma \), by \(|\gamma |\), its interior (the sites in \({\mathbb {Z}}^2\) it surrounds) by \(\Lambda _\gamma \) and its interior area (the number of such sites) by \(|\Lambda _\gamma |\). Moreover we let \(\Delta _{\gamma }\) be the set of sites in \({\mathbb {Z}}^2\) such that either their distance (in \({\mathbb {R}}^2\)) from \(\gamma \) is \(\tfrac{1}{2}\), or their distance from the set of vertices in \({{\mathbb {Z}}^2}^*\) where two non-linked bonds of \(\gamma \) meet equals \(1{/}\sqrt{2}\). Finally we let \(\Delta ^+_\gamma =\Delta _\gamma \cap \Lambda _\gamma \) and \(\Delta ^-_\gamma = \Delta _\gamma {\setminus } \Delta ^+_\gamma \). Given a contour \(\gamma \) we say that \(\gamma \) is an (h-contour) for the configuration \(\eta \) if

Finally \({\mathscr {C}}_{\gamma ,h}\) will denote the event that \(\gamma \) is an h-contour.
To illustrate the above definitions with a simple example, consider the elementary contour given by the square of side 1 surrounding a site \(x\in {\mathbb {Z}}^2\). In this case, \(\gamma \) is an h-contour iff \(\eta ({x})\geqslant h\) and \(\eta (y)\leqslant h-1\) for all \(y\in \{x\pm e_1, x\pm e_2, x+ e_1+e_2,x-e_1-e_2\}\). We observe that a geometric contour \(\gamma \) could be at the same time a h-contour and a \(h'\)-contour with \(h\ne h'\). More generally two geometric contours \(\gamma ,\gamma '\) could be contours for the same surface with different height parameters even if \(\gamma \cap \gamma '\ne \emptyset \), but then the interior of one of them must be contained in the interior of the other; see Fig. 1 for an example.
Example of a SOS configuration above the wall in the \(7\times 7\) box \(\Lambda _3\) with zero boundary conditions: white sites have height 0, shaded sites have height 1 and darker sites have height 2. Notice that according to Definition 2.1 there are three 1-contours and two 2-contours
2.3 Cluster expansion
So called cluster expansions are a well established tool for the analysis of random interfaces at low-temperature; see e.g. [3] where both the SOS model and the discrete gaussian model are considered. Here we shall need a particular expansion that allows us to take into account the extra constraints that appear naturally in the partition function of the SOS model on a region \(\Lambda \) delimited by two contours; see (2.5) below.
Given a finite connected set \(\Lambda \subset {\mathbb {Z}}^2\), let \(\partial _*\Lambda \) denote the set of \(y\in \Lambda \) either at distance 1 from \(\partial \Lambda \) or at distance \(\sqrt{2}\) from \(\partial \Lambda \) in the south-west or north-east direction. Fix \(U_+,U_-\subset \partial _* \Lambda \), and let \( Z_{\Lambda ,U_+,U_-}\) denote the SOS partition function in \(\Lambda \) with the sum over \(\eta \) restricted to those \(\eta \in \Omega _\Lambda ^0\) such that \(\eta (x)\geqslant 0\) for all \(x\in U_+\) and \(\eta (x)\leqslant 0\) for all \(x\in U_-\). Clearly, if \(U_-\cap U_+\ne \emptyset \), then \(\eta (x)=0\) is fixed for all \(x\in U_-\cap U_+\). If \(\Lambda =\emptyset \) then \(Z_{\Lambda ,U_+,U_-}:= 1\). We refer the reader to [5, App. A] for a proof of the following statement based on the general cluster expansion from [13].
Lemma 2.2
There exists \(\beta _0>0\) independent of \(\Lambda \) such that for all \(\beta \geqslant \beta _0\), for all finite connected \(\Lambda \subset {\mathbb {Z}}^2\) and \(U_+,U_-\subset \partial _* \Lambda \):
where the potentials \(\varphi _{U_+,U_-}(V)\) satisfy
-
(i)
\(\varphi _{U_+,U_-}(V)=0\) if V is not connected.
-
(ii)
\(\varphi _{U_+,U_-}(V)=\varphi _0(V)\) if \(V\cap ({U_+\cup U_-})=\emptyset \), for some shift invariant potential \(V\mapsto \varphi _0(V)\), that is
$$\begin{aligned} \varphi _0(V)= \varphi _0(V+x)\quad \forall \,x\in {\mathbb {Z}}^2, \end{aligned}$$where \(\varphi _0\) is independent of \(U_+,U_-,\Lambda \).
-
(iii)
For all \(V\subset \Lambda \):
$$\begin{aligned} \sup _{{U_+,U_-}\subset \partial _* \Lambda }|\varphi _{U_+,U_-}(V)|\leqslant \exp (-(\beta -\beta _0)\, d(V)) \end{aligned}$$where d(V) is the cardinality of the smallest connected set of all dual bonds separating points of V from points of its complement (a dual bond separates V from \(V^c\) iff it is orthogonal to a bond connecting V to \(V^c\)).
2.4 Nested contours
Consider the rectangle \(\Lambda _{L,M}\), for some \(L,M\in {\mathbb {N}}\), and let \({\mathbb {P}}_{\Lambda }\) denote the SOS Gibbs measure in \(\Lambda :=\Lambda _{L,M}\) with zero boundary conditions. Given two contours \(\gamma ,\gamma '\), we write \(\gamma \subset \gamma '\) if \(\Lambda _\gamma \subset \Lambda _{\gamma '}\). Fix \(n\in {\mathbb {N}}\) and pick n geometric contours \(\gamma _1,\ldots ,\gamma _n\) such that \( \gamma _{i+1}\subset \gamma _i\), for every \(i=1,\ldots ,n-1\). Consider the event \(\cap _{i=1}^n{\mathscr {C}}_{\gamma _i,i}\) that \(\gamma _i\) is an i-contour for all \(i=1,\ldots ,n\). The probability of this event under \({\mathbb {P}}_{\Lambda }\) can be expressed as
where \(Z_\Lambda \) denotes the partition function of the SOS model in \(\Lambda =\Lambda _{L,M}\) with zero boundary conditions and \( Z(\gamma _1,\ldots ,\gamma _n; L,M)\) stands for the same summation restricted to the configurations \(\eta \in \Omega _\Lambda ^0\) such that \(\gamma _i\) is an i-contour for each \(i=1,\ldots ,n\). By applying the cluster expansion in Lemma 2.2, with \(\Lambda =\Lambda _{L,M}\) and \(U_\pm =\emptyset \), we can write
To expand the partition function \(Z(\gamma _1,\ldots ,\gamma _n; L,M)\), define \(S_i:=\Lambda _{\gamma _{i-1}}{\setminus } \Lambda _{\gamma _i}\), for \(i=1,\ldots ,n+1\), where \(\Lambda _{\gamma _{0}}=\Lambda \) and \(\Lambda _{\gamma _{n+1}} =\emptyset \), and set \(\Delta _i^+=S_i\cap \Delta ^+_{\gamma _{i-1}}\), and \(\Delta _i^-=S_i\cap \Delta ^-_{\gamma _{i}}\), with the understanding that \(\Delta _1^+ = \Delta _{n+1}^- =\emptyset \). Notice that \(\Delta _i^\pm \subset \partial _*S_i\). Using the notation of Lemma 2.2 a direct computation proves that
The term \(\sum _{i=1}^n|\gamma _i|\) accounts for the minimal energy associated to the given contours. The fact that the surface gradient across a contour \(\gamma _i\) must be at least 1 is encoded by the constraints on \(\Delta _i^+,\Delta _i^-\) appearing in \(Z_{S_i,\Delta _i^+,\Delta _i^-}\).
Therefore, the expansion (2.2) implies
The ratio (2.3) then becomes
where
where the condition \(V\cap (\cup _{i=1}^n\gamma _i) \ne \emptyset \) means that V intersects more than just one \(S_i\). When \(n=1\), we have only one contour \(\gamma _1=\gamma \) and we define
Observe that property (iii) of the potentials \(\varphi _{U_+,U_-}(V)\) in Lemma 2.2 implies in particular that
where \(\lim _{\beta \rightarrow \infty }\varepsilon (\beta )=0\). Later on it will be convenient to introduce the quantity \(\psi _\infty (\gamma )\) defined as \(\psi _\Lambda (\gamma )\) but without the restriction \(V\subset \Lambda \), i.e. now \(S_1= {\mathbb {Z}}^d{\setminus }\Lambda _\gamma \).
2.5 The staircase ensemble
Consider the rectangle \(\Lambda _{L,M}\), for some \(L,M\in {\mathbb {N}}\). Fix \(n\in {\mathbb {N}}\) and integers
and set \(a_0=b_0=-(M+1)\) and \(a_{n+1}=b_{n+1}=M+1\). We define a “staircase” height \({\tau }\) at the external boundary \(\partial \Lambda _{L,M}\) of our rectangle which, starting from height zero at the base of the rectangle (i.e. the set \((x,-(M+1)), x=-L,\ldots ,L\)) jumps by one at the locations specified by the two n-tuples \(\{a_i,b_i\}\) until it reaches height n:
where \(i\in \{0,\ldots ,n\}\), see Figure 2. Note that if two or more values of the \(a_i\) or \(b_i\) coincide then the boundary height \({\tau }\) takes jumps higher than 1 at those points.
Next, let \(Z(a_1,\ldots ,a_n; b_1,\ldots , b_n; L,M)\) denote the partition function of the SOS model in \(\Lambda _{L,M}\) with boundary condition \({\tau }\) as in (2.11).
A sketch of the staircase boundary condition (2.11) in the rectangle \(\Lambda _{L,M}\) for \(n=2\). The points \(z_i,z'_i\) have coordinates \(z_i=(-L-1,a_i)\), and \(z'_i=(L+1,b_i)\)
Let also \(Z_\Lambda \) denote as above the partition function of the SOS model in \(\Lambda =\Lambda _{L,M}\) with zero boundary condition everywhere. We want to compute the ratio
To expand the partition function in the numerator of (2.12), we need the notion of an open contour. This is defined as in Definition 2.1 except that \(e_0\ne e_n\). Since the boundary conditions force the surface height to grow from height zero at the bottom base of \(\Lambda _{L,M}\) to height n at the top base, necessarily any configuration \(\eta \) of the SOS interface compatible with \(\tau \) must satisfy the following property.
Given \(\eta \) there exist uniquely defined non-crossing open contours \(\gamma _i\), \(i=1,\ldots , n\), joining the dual lattice points \(x_i:=(-L-1/2,a_i-1/2)\) and \( y_i:=(L+1/2,b_i-1/2)\) such that \(\eta (x)\leqslant i-1\) for all \(x\in \Delta _i^-\) and \(\eta (x)\geqslant i-1\) for all \(x\in \Delta _i^+\) where \(\Delta _i^\pm \) are now the sets defined as follows. Let \(S_i\subset \Lambda _{L,M}\) denote the region bounded by \(\gamma _i\) and \(\gamma _{i-1}\), where \(\gamma _{n+1}\) is the top boundary of \(\Lambda _{L,M}\) and \( \gamma _{0}\) is the bottom boundary of \(\Lambda _{L,M}\). Then \(\Delta _i^-\) is defined as the set of \(x\in S_i\) such that either their distance from \(\gamma _{i}\) is \(\tfrac{1}{2}\), or their distance from the set of vertices in \({{\mathbb {Z}}^2}^*\) where two non-linked bonds of \(\gamma _{i}\) meet equals \(1/\sqrt{2}\). Similarly, \(\Delta _i^+\) is the set of \(x\in S_i\) such that either their distance from \(\gamma _{i-1}\) is \(\tfrac{1}{2}\), or their distance from the set of vertices in \({{\mathbb {Z}}^2}^*\) where two non-linked bonds of \(\gamma _{i-1}\) meet equals \(1/\sqrt{2}\). Lemma 2.2 here implies
where the sum ranges over all possible values of the open contours \(\gamma _i:x_i\rightarrow y_i\) inside \(\Lambda _{L,M}\) with the non-crossing constraints. Recalling that \(Z_\Lambda \) can be expanded as in (2.4), one finds that
where
where the condition \(V\cap (\cup _{i=1}^N\gamma _i) \ne \emptyset \) means that V intersects more than just one \(S_i\). Equation (2.13) expresses the ratio (2.12) as the partition function of a gas of n interacting non-crossing open contours \(\gamma _1,\ldots ,\gamma _n\) within \(\Lambda _{L,M}\) such that \(\gamma _i:x_i\rightarrow y_i\), \(i=1,\ldots ,n\). Using (iii) in Lemma 2.2 the limit as \(M\rightarrow \infty \) of the above expression is well defined, so that the following holds.
Lemma 2.3
For any integers n, \(\{a_i,b_i\}_{i=1}^n\) satisfying (2.10), the limit
exists and it satisfies
where the sum ranges over all possible values of the open contours in the strip \(\Lambda _{L,\infty }:=[-L,L]\times {\mathbb {Z}}\) and \(\Phi _{L,\infty }\) is defined as in (2.14) with \(\Lambda _{L,M}\) replaced by \(\Lambda _{L,\infty }\).
Proof
Using (iii) in Lemma 2.2 it is immediate to check that for any family of contours \((\gamma _1,\ldots ,\gamma _n)\)
As in (2.9) we have
Hence, for \(\beta \) large enough, the conclusion follows by dominated convergence. \(\square \)
2.6 Surface tension
Here we recall the definition and some properties of the surface tension corresponding to arbitrary tilt. Let us rewrite (2.15) in the case \(n=1\) as
where the sum ranges over all open contours in the strip \(\Lambda _{L,\infty }\) joining the dual lattice points \(x_1:=(-L-1/2,a_1-1/2)\) and \( y_1:=(L+1/2,b_1-1/2)\).
Lemma 2.4
There exists \(\beta _0>0\) such that the following holds for all \(\beta \geqslant \beta _0\). Let \({\mathcal {Z}}(a_1;\, b_1;\, L)\), denote the partition function (2.17). Assume that as \(L\rightarrow \infty \) one has \((b_1-a_1)/(2L)\rightarrow \lambda \in {\mathbb {R}}\) and set \(\theta =\tan ^{-1}(\lambda )\). Then the function
is well defined and positive in \((-\pi /2,\pi /2)\). It is convex in the following sense: defining, for \(x\in {\mathbb {R}}^2\), \(\tau _\beta (x)=\Vert x\Vert \tau _\beta (\theta _x)\) with \(\theta _x\) the angle formed by the vector x with the horizontal axis, \(\tau _\beta \) is a convex function on \({\mathbb {R}}^2\). Moreover,
Proof
Existence and the stated properties of the surface tension are known facts [11, Section 4.16]. It is also known, see [11, Section 4.20], that \(\tau _\beta (\theta )\) tends to \(|\cos \theta \,|+ |\sin \theta \,|\), as \(\beta \rightarrow \infty \). In particular, \({\tau }_\beta (0)\rightarrow 1\), \(\beta \rightarrow \infty \). Strictly speaking the proofs in [11] are carried out for the contour ensemble associated to the 2D Ising model, which has the form (2.17) but with slightly different potentials in the “decoration term” \(\Phi _{L,\infty }(\gamma )\). However, thanks to the properties listed in Lemma 2.2, the same proofs actually apply to our model in (2.17) as well.
To prove (2.18) we distinguish two cases. If \(|b_1-a_1|>4L \) we use again (2.9) to obtain
where \(\gamma _1\) is a contour from \((-(L+1,a_1)\) to \(((L+1),b_1)\). Clearly the above sum is negligible w.r.t. \(\exp (-2\beta L\tau _\beta (0))\) as \(L\rightarrow \infty \) for \(\beta \) large enough. If instead \(|b_1-a_1|\leqslant 4L\), then the estimate [11, Eq. (4.12.3)] together with convexity of the surface tension allows one to conclude (2.18). \(\square \)
It is not hard to check that the special case \(\theta =0\) coincides with the quantity in Definition 1.1. Indeed, using the notation from Definition 1.1 together with (2.13) (with \(n=1\) and \(a_1=b_1=0\)),
The same arguments of Lemma 2.3 can be used to check that
3 Lower bound
Here we prove the lower bound in Theorem 1.2. We first establish a lower bound on the probability of having zero height at the boundary of a square.
Lemma 3.1
For \(\beta \geqslant \beta _0\) there exists \(c_\beta >0\) such that for any \(L\in {\mathbb {N}}\):
Proof
Recall that \({\mathbb {P}}(\cdot )=\lim _{K\rightarrow \infty }{\mathbb {P}}_{\Lambda _K}(\cdot )\). Expanding as in (2.4), we see that
where \(V\cap \partial \Lambda _L\ne \emptyset \) is equivalent to V not contained in \(\Lambda _K{\setminus } \partial \Lambda _L\). From the decay properties of the potentials \(\varphi _0\) stated in Lemma 2.2, the desired result follows. \(\square \)
3.1 Proof of the lower bound in Theorem 1.2
If we prove the lower bound for \({\mathbb {P}}_{\Lambda _L}\) in (1.2) we also have the same lower bound for \({\mathbb {P}}\) by using Lemma 3.1 and
To prove the lower bound for \({\mathbb {P}}_{\Lambda _L}\) we proceed by restricting the set of configurations to an event E defined as follows. Fix \( N:=H(L)=\lfloor \tfrac{1}{4\beta } \log L\rfloor \). Define the nested annular regions \(\bar{\mathcal {U}}_i:=\Lambda _{L-3\ell _{i-1}}{\setminus }\Lambda _{L-3\ell _i}\), \(i=1,\ldots , N\), where \(\ell _0=0\) and \(\ell _i=i(i+1)/2\). Notice that each \(\bar{\mathcal {U}}_i\) consists of 3 nested disjoint annuli each of width i. We define \({\mathcal {U}}_i\) as the middle one, i.e. \({\mathcal {U}}_i = \Lambda _{L-(3\ell _{i-1} -i)}{\setminus }\Lambda _{L-(3\ell _i+i)}\). These sets are such that \(d({\mathcal {U}}_i,{\mathcal {U}}_{i+1})\geqslant 2i +1\), where \(d(\cdot ,\cdot )\) stands for the euclidean distance.
For each i, define the set \({\mathcal {C}}_i\) of all contours \(\gamma \) such that \(\gamma \subset {\mathcal {U}}_i\) and \(\gamma _i\) surrounds \(\Lambda _{L-(3\ell _i+i)}\). We consider the event E that for each \(i=1,\ldots ,N\) there exists an i-contour \(\gamma _i\in {\mathcal {C}}_i\):
For a fixed choice of \(\gamma _i\subset {\mathcal {C}}_i\), \(i=1,\ldots , N\) we write \(S_i=\Lambda _{\gamma _{i-1}}{\setminus } \Lambda _{\gamma _i}\), and \(\Delta _i^+=S_i\cap \Delta ^+_{\gamma _{i-1}}\), and \(\Delta _i^-=S_i\cap \Delta ^-_{\gamma _{i}}\) as in Sect. 2.4. We define \(Z_{S_i,\Delta _i^+,\Delta _i^-}^+\) as the partition function in \(S_i\) with boundary conditions \(i-1\) in \(\partial S_i\), and with the following constraints: \(\eta (x)\leqslant i-1\) for \(x\in \Delta _i^+\), \(\eta (x)\geqslant i-1\) for \(x\in \Delta _i^-\) and \(\eta (x)\geqslant 0\) for all \(x\in S_i\). Then
Below, we shall take \(n:=\lfloor \log \log L\rfloor \) and fix arbitrary contours \(\gamma ^*_1\in {\mathcal {C}}_1,\ldots ,\gamma ^*_n\in {\mathcal {C}}_n\), and sum over the remaining contours \(\gamma _i\), \(i=n+1,\ldots ,N\)
Lemma 3.2
Fix \(\beta \geqslant \beta _0\) and fix \(\gamma ^*_1\in {\mathcal {C}}_1,\ldots ,\gamma ^*_n\in {\mathcal {C}}_n\), where \(n=\lfloor \log \log L\rfloor \). Then
Proof
Let \(F_i\) denote the event that there is more than one i-contour in \({\mathcal {C}}_i\). Then
Thus, it suffices to show that for any fixed choice of \(\gamma ^*_k\in {\mathcal {C}}_k\), \(k=1,\ldots ,n\) and \(\gamma _j\subset {\mathcal {C}}_j\), \(j=n+1,\ldots , N\):
Suppose the j-contour \(\gamma _j\in {\mathcal {C}}_j\) is given for each \(j=n+1,\ldots ,N\). If \(F_i\) occurs then there must be a i-contour \(\gamma \in {\mathcal {C}}_i\), \(\gamma \ne \gamma _i\), such that either \(\gamma \subset S_{i+1}\) or \(\gamma \subset S_{i}\). In particular, if \(\cup _{i=n+1}^NF_i\) occurs, then, for some \(i\in [ n+1,N+1]\), there exists either an \((i-1)\)-contour or an i-contour \(\gamma \) inside \(S_i\) and surrounding \(\Lambda _{L-(3\ell _i+i)}\). Let \(\pi _{S_i,\Delta _i^+,\Delta _i^-}^+\) denote the probability measure corresponding to the partition function \(Z_{S_i,\Delta _i^+,\Delta _i^-}^+\). From [7, Proposition 2.7] one has that for any fixed contour \(\gamma \) inside \(S_i\), for any \(h\in {\mathbb {N}}\):
Here and below, by C we mean a positive constant that does not depend on \(\beta \) and L, whose value may change at each occurrence. Since \(|S_i| \leqslant CL i\leqslant L\log L\), and \(\log |\gamma |\leqslant 2\log L\), taking either \(h=i\) or \(h=i-1\), with \(i\geqslant n+1\) one has that \(e^{-4\beta h}|S_i|\leqslant L(\log L)^{1-4\beta }\) and \(e^{-4\beta h}\log |\gamma |\leqslant 2(\log L)^{1-4\beta }\), and therefore
as soon as \(\beta \) and L are large enough. If \(\gamma \) is required to surround \(\Lambda _{L-3\ell _i+i}\), then necessarily \(|\gamma |\geqslant 2L\). Let \(p_i\) denote the \(\pi _{S_i,\Delta _i^+,\Delta _i^-}^+\)-probability that there exists either an \((i-1)\)-contour or an i-contour \(\gamma \) inside \(S_i\) and surrounding \(\Lambda _{L-3\ell _i+i}\). Summing over \(\gamma \subset S_i\) with \(|\gamma |\geqslant 2L\) in (3.3), one finds that for \(\beta \) large enough, \(p_i\leqslant e^{-L}\). From (3.2), using a union bound and the fact that \(Ne^{-L}\leqslant 1/2\), it follows that
\(\square \)
Thanks to Lemma 3.2 the lower bound in Theorem 1.2 follows if we prove that
for any fixed choice of \(\gamma ^*_k\in {\mathcal {C}}_k\), \(k=1,\ldots ,n\), with \(n=\lfloor \log \log L\rfloor \). To prove (3.4) we start by observing that by the FKG inequality one has
where \(Z_{S_i,\Delta _i^+,\Delta _i^-}^+\) is defined above (3.2), \(Z_{S_i,\Delta _i^+,\Delta _i^-}\) is as in Sect. 2.3, and \(\pi _{S_i,\Delta _i^+,\Delta _i^-}\) denotes the probability measure associated to the partition function \(Z_{S_i,\Delta _i^+,\Delta _i^-}\). From [5, Proposition 3.9] one has that \(\pi _{S_i,\Delta _i^+,\Delta _i^-}(\eta (x)\geqslant 0)\geqslant 1-Ce^{-4\beta (i-1)}\) for any \(x\in S_i\). Using \(1-x\geqslant e^{-2x}\) for \(0\leqslant x\leqslant 1/2\), one has
Therefore, in (3.2) we can estimate
Expanding as in (2.6) one obtains
where \(\Psi \) is given in (2.7). Estimating \(|S_i|\leqslant Ci L\) one finds
On the other hand, the term \(|S_{N+1}|e^{-4\beta N} = |\Lambda _{\gamma _N}|e^{-4\beta H(L)}\) satisfies
where we use \(e^{-4\beta H(L)}\leqslant C/L\). Note that it is at this point of the argument that it is crucial to have N as large as H(L). From (3.6)-(3.7) one has \(\sum _{i=1}^{N+1} |S_i|e^{-4\beta (i-1)}\leqslant CL\). From this bound and (3.5) we obtain
Next, we want to show that the interactions among the contours are negligible in our setting. Let \(\psi _\Lambda (\gamma )\) denote the potential associated to a single contour \(\gamma \) as defined in (2.8).
Lemma 3.3
Take \(\beta \geqslant \beta _0\). Uniformly in the choice of \(\gamma _1\in {\mathcal {C}}_1,\ldots ,\gamma _N\in {\mathcal {C}}_N\) one has
Proof
Notice that any \(V\subset \Lambda \) such that \(d(V,\gamma _i)\leqslant 1\) and \(d(V,\gamma _{i+1})\leqslant 1\) must have \(d(V)\geqslant 2i\). Thus the sum of the potentials associated to V’s that have \(d(V,\gamma _i)\leqslant 1\) and are such that \(d(V,\gamma _j)\leqslant 1\) for some \(j\ne i\) contributes at most \(|\gamma _i|e^{-\beta i/2}\) if \(\beta \) is large enough. \(\square \)
From (3.8) and Lemma 3.3 one has
For \(i=1,\ldots ,n\), we can use the rough estimates \(|\gamma _i|\leqslant CL n \leqslant C L \log \log L\) and \(|\psi _\Lambda (\gamma _i)| \leqslant C|\gamma _i|\) (cf. (2.9)) to obtain
For \(n<i\leqslant N\) we need the following statement.
Lemma 3.4
Uniformly over i such that \(n<i\leqslant N\), one has
We first conclude the proof of the lower bound in Theorem 1.2, assuming the estimate of Lemma 3.4. From Lemma 3.2 and (3.9)–(3.10) we have
From Lemma 3.4 and using \(NL=1/(4\beta )L\log L+O(L)\) one has
This concludes the proof.
Proof of Lemma 3.4
First observe that \(\gamma \in {\mathcal {C}}_i\) implies \(|\gamma |\leqslant |S_i|\leqslant L\log L\) and therefore for \(i\geqslant \log \log L\) and \(\beta \geqslant \beta _0\) one has
Next, observe that we may safely replace \(\psi _\Lambda (\gamma )\) in (3.11) by the quantity \(\psi _\infty (\gamma )\) (see the end of Sect. 2.4). Indeed, any connected set V that touches both \({\mathcal {U}}_i\) and \(\partial \Lambda \) must have \(d(V)\geqslant \tfrac{1}{2}(\log \log L)^2\). Thus, we have to show that
To prove (3.12) we fix i and partition the set \({\mathcal {U}}_i\) into rectangles \(R_j\), \(j=1,\ldots ,m\), with height i and basis \(i^{2-\varepsilon }\), so that there are \(m\sim 8Li^{-2+\varepsilon }\) such rectangles, see Fig. 3. For simplicity, let us assume that the partitioning is exact so that \({\mathcal {U}}_i\) is the union of the \(R_j\)’s plus four squares at the corners as in Fig. 3. The modifications in the general case are straightforward.
We fix for every rectangle \(R_j\) the points \(x_j\) and \(y_j\) that are the midpoints of the two shorter side. Consider an open contour \({\hat{\gamma }}_j\) connecting \(x_j\) to \(y_j\) which is entirely contained in \(R_j\) (see Fig. 3). For technical reasons it is convenient to consider a closed path \({\hat{\gamma }}\) that agrees with \({\hat{\gamma }}_j\) on \(R_j\). The latter is defined as follows. Let \(\hat{\gamma }\) be the closed contour contained in \({\mathcal {U}}_i\) which coincides with \({\hat{\gamma }}_j\) inside \(R_j\), it is given by straight segments in all other rectangles \(R_k\), \(k\ne j\), and by a straight right angle shape at each of the four corner squares. Then we define \(\psi _\infty ({\hat{\gamma }}_j)\) as \(\psi _\infty ({\hat{\gamma }})\) (see text after (2.9)) but with the restriction to those sets V which have distance from \({\hat{\gamma }}_j\) at most 1. It follows from [11, Sections 4.12 and 4.15] that for a fixed index j one has, for i large:
The point is that the height i of the rectangles \(R_j\) is much larger than the typical vertical fluctuation \(i^{1-\varepsilon /2}\) of paths \({\hat{\gamma }}_j\), so the restriction to be in \(R_j\) is not modifying the partition function significantly.
Suppose now that \(\gamma \in {\mathcal {C}}_i\) is a contour passing through all the points \(x_j,y_j \) that can be written as the composition of \({\hat{\gamma }}_1,\ldots ,{\hat{\gamma }}_m\) where \( {\hat{\gamma }}_j\) is as in the sum above, and assume that it has some prescribed shape at the four corners of \({\mathcal {U}}_i\), e.g. a right angle form as in Fig. 3. Then it is immediate to check that \(|\gamma |\leqslant \sum _{j=1}^m|{\hat{\gamma }}_j| + O(i) \), and
The latter estimate holds thanks to the decay properties of the potentials, so that the mutual interaction between \({\hat{\gamma }}_j\) and \({\hat{\gamma }}_{j-1}\) is O(i) uniformly in \(j=1,\ldots , m\). Thus, by restricting the sum in (3.12) to contours as in (3.13) one obtains
Since \(m\sim 8Li^{-2+\varepsilon }\), the desired estimate follows. \(\square \)
4 A monotonicity property of the SOS model
Recall the staircase ensemble defined in Sect. 2.5 with partition function
as defined in Lemma 2.3. In this section we establish the following important monotonicity property.
Theorem 4.1
There exists \(\beta _0>0\) such that, for any \(\beta >\beta _0\) and any \(L\in {\mathbb {N}}\)
The above estimate allows one to control the partition function of n interacting open contours by means of the partition functions of n non-interacting open contours. In particular, Theorem 4.1 and Lemma 2.4 yield the following corollary.
Corollary 4.2
Fix \(n\in {\mathbb {N}}\), and suppose that as \(L\rightarrow \infty \) one has \((b_i-a_i)/L\rightarrow \lambda _i \in {\mathbb {R}}\), \(i=1,\ldots ,n\). Then
where \(\theta _i=\tan ^{-1}(\lambda _i)\).
The proof of Theorem 4.1 is based on the following key lemma.
Lemma 4.3
Given \(\{a_i,b_i\}_{i=1}^n\), let \(\{a'_i,b'_i\}_{i=1}^n\) be defined by
Then
Proof of Lemma 4.3
Set \(\Lambda := \Lambda _{L,M}\) for some large fixed \(M>\max \{a_n,b_n,-a_1,-b_1\}\). Let \({\tau },{\tau }'\) be the SOS boundary conditions associated to \(\{a_i,b_i\}_{i=1}^n\) and \(\{a'_i,b'_i\}_{i=1}^n\) according to (2.11). Given \(s\in [0,1]\) consider the auxiliary boundary condition \(\tau _s: \partial \Lambda \mapsto {\mathbb {R}}\) defined by
Next, we consider the partition function \(Z_\Lambda ^{\tau _s}\) associated to \({\tau }_s\) (strictly speaking we have only defined the model for integer valued boundary condition, but it is straightforward to extend it to the real valued case). Notice that \({\tau }_s = s{\tau }+(1-s){\tau }'\). We shall see that \(Z_\Lambda ^{\tau _s}\) is differentiable w.r.t. \(s\in [0,1]\) so that
In order to compute the above derivative we proceed as follows. Define the points \(z=(-(L+1),a_n),w=(-L,a_n)\) and \(z'=(L+1,b_n),\ w'=(L,b_n)\), so that w (resp. \(w'\)) is the nearest neighbor of z (resp. \(z'\)) in \(\Lambda \), see Fig. 4.
A sketch of the staircase boundary condition with \(n=2\) steps as seen from above, with two open contours and the pairs of vertices appearing in the proof of Lemma 4.3: \(z=(-L-1,a_2), w=(-L,a_2)\), \(z'=(L+1,b_2), w'=(L+1,b_2)\)
Let \({\mathcal {B}}_\Lambda ^*={\mathcal {B}}_\Lambda {\setminus }\{wz,w'z'\}\) denote all bonds with at least one vertex in \(\Lambda \) with the exception of the two bonds wz and \(w'z'\). Define the energy function \({\mathcal {H}}_\Lambda ^{{\tau },*}(\eta ),\ \eta \in \Omega _\Lambda ^{{\tau }_s}\) by
where
Since the bonds wz and \(w'z'\) are not included in the above sum, we see that \({\mathcal {H}}_\Lambda ^{{\tau },*}(\eta )\) does not depend on the parameter s. Let also
Define the partition function \(\Xi ^{{\tau },*}_\Lambda = \sum _{\eta \in \Omega _\Lambda ^{{\tau }_s}}\exp (-\beta {\mathcal {H}}_\Lambda ^{{\tau },*}(\eta ))\), and the Gibbs measure
\(\eta \in \Omega _\Lambda ^{{\tau }_s}\). It is not hard to check that
Using the above expression for \(Z_\Lambda ^{\tau _s}\) we get
where, for any \(s\in [0,1]\), we define
The function \(G_{s,n}\) takes values in \(\{-2e^{-2\beta s},0,2e^{-2\beta (1-s)}\}\) and is easily seen to be increasing in the configuration \(\eta \). Therefore, if we raise to height \(n-1\) the value of \({\tau }\) on those boundary vertices where it was at most \(n-1\) and we denote by \({\hat{{\tau }}}\) the resulting boundary condition, from the FKG inequality we get that
The validity of the FKG inequality follows from lattice condition (2.1) for the measure \(\pi _\Lambda ^{{\tau },*}\), which can be verified directly. The boundary height \({\hat{{\tau }}}\) has now a single step from level \(n-1\) to level n. Using vertical translation invariance we can now safely replace the height of \({\hat{{\tau }}}\) by 0, 1 instead of \(n-1,n\). Finally, since \(G_{s,1}\) is a bounded local function, we can take the limit \(M\rightarrow \infty \) in (4.2) and get that
where \(\pi ^{{\hat{{\tau }}},*}_\infty (\cdot )\) denotes the weak limit as \(M\rightarrow \infty \) of \(\pi _\Lambda ^{{\hat{{\tau }}},*}\), that is the Gibbs measure on \(\Lambda _{L,\infty }=[-L,L]\times {\mathbb {Z}}\) with boundary condition at height 1 at the vertices \(x=(x_1,x_2)\) with either \(x_1=-(L+1)\) and \(x_2\geqslant a_n+1\) or \(x_1=L+1\) and \(x_2\geqslant b_n+1\); the boundary height is unspecified at the vertices \(z,z'\) (this simply means that the terms corresponding to bonds wz and \(w'z'\) do not appear in the interaction) and otherwise it is equal to zero. The existence of the limits mentioned above can be proved again from the cluster expansion representation as in Lemma 2.3. By symmetry one has that
so that
In conclusion
and the lemma is proved.\(\square \)
We can now complete the proof of Theorem 4.1. By iterating Lemma 4.3 arbitrarily many times, we have that
On the other hand, using the explicit representation (2.15) together with the rough bound (2.16) to control the large deviations of the n-th contour \(\gamma _n\), we have that
In conclusion, we have factorized out the contribution of the n-th contour. By repeating the above reasoning for \((a_{n-1},b_{n-1}),\,(a_{n-2},b_{n-2})\ldots ,(a_{2},b_{2})\) we finally get (4.1).
5 Upper bound
If we prove the upper bound for \({\mathbb {P}}\) in (1.2), then we can obtain the upper bound for \({\mathbb {P}}_{\Lambda _L}\) by using (3.1) and Lemma 3.1. From now on we concentrate on proving the upper bound for \({\mathbb {P}}\).
For any event A, note that
Indeed, (5.1) is obtained by multiplying by \({\mathbb {P}}( \eta _{\Lambda _L}\geqslant 0)\) both sides of the obvious inequality \(1\leqslant {\mathbb {P}}(A)/ {\mathbb {P}}(A,\eta _{\Lambda _L}\geqslant 0)\).
For any \(\delta >0\) and \(K>0\), define \(A=A(\delta ,K)\), as the event that there exists a lattice circuit \({\mathcal {C}}\) surrounding \(\Lambda ':=\Lambda _{(1-\delta )L}\) such that \(\eta (x)\geqslant H(L) - K\), for all \(x\in {\mathcal {C}}\), where as usual \(H(L)=\lfloor \tfrac{1}{4\beta } \log L\rfloor \).
Proposition 5.1
For any \(\delta >0\), there exists a constant \(K>0\) such that
Proof
Let \(\partial _* \Lambda _L\) denote the internal boundary of \(\Lambda _L\). Observe that \(A(\delta ,K)\) is monotone increasing so that by the FKG inequality
Therefore, the proposition follows once we know that for some \(K=K(\delta )\) one has
Under the conditioning \(\eta _{\Lambda _L}\geqslant 0, \,{\eta _{\partial _*\Lambda _{L}}}=0\), one has an SOS interface in \(\Lambda _{L-1}\) with a wall at height zero and zero boundary conditions. The result of [7, Theorem 2] implies that with probability converging to 1, within \(\Lambda _{L-1}\), there exists an h-contour surrounding \(\Lambda '=\Lambda _{(1-\delta )L}\), for all \(h\leqslant H(L) -K\) as soon as K is a sufficiently large constant depending on \(\delta \). This implies (5.2). \(\square \)
It follows that to prove the upper bound in (1.2) it is sufficient to establish:
Proposition 5.2
For any \(\delta >0\), for any \(K>0\), one has
5.1 Proof of Proposition 5.2
The first observation is that we may impose zero boundary conditions outside a very large set, e.g. \(\Lambda _M\) with \(M\gg L^2\), and therefore we may consider \(\widetilde{\mathbb {P}}:={\mathbb {P}}_{\Lambda _M}\) instead of \({\mathbb {P}}\) in (5.3). The reason is that the probability that there is a contour surrounding \(\Lambda '\) and not contained in, say, \(\Lambda _{L^2}\) is a negligible \(O(\exp (-L^2))\), as one can check easily using a rough estimate as in (2.9). Then, \(A(\delta ,K)\) can be considered as a local event (localized in \(\Lambda _{L^2}\)) and by definition of thermodynamic limit one can approximate arbitrarily well \({\mathbb {P}}( A(\delta ,K)) \) by \(\widetilde{\mathbb {P}}( A(\delta ,K)) \), if M is sufficiently large.
The event \(A(\delta ,K)\) implies that for each \(h=1,\ldots ,N:=H(L) - K\) there exists (at least) one h-contour surrounding \(\Lambda '\). Therefore, there must exist \(\Lambda _M\supset \gamma _1\supset \cdots \supset \gamma _N\supset \Lambda '\) such that \(\gamma _h\) is an h-contour:
Here we use the notation \(\Lambda _M\supset \gamma _1\supset \cdots \supset \gamma _N\supset \Lambda '\) when the contours satisfy \(\Lambda _M\supset \Lambda _{\gamma _1}\supset \cdots \supset \Lambda _{\gamma _N}\supset \Lambda '\).
For a fixed choice of \(\gamma _1\supset \cdots \supset \gamma _N\) the above probability is computed in (2.6):
To deal with the summation in (5.4) we consider a decomposition of each contour into four “irreducible” pieces, which will be responsible for the main contributions, plus some negligible corner terms.
Let \({\mathcal {S}}_v\) and \({\mathcal {S}}_h\) denote, respectively, the vertical and horizontal infinite strips obtained by prolonging the sides of the square \(\Lambda '\):
Let \({\mathcal {S}}_v^t\), resp. \({\mathcal {S}}_v^b\), denote the top, resp. bottom part of \({\mathcal {S}}_v\), i.e. the part that lies above, resp. below, the square \(\Lambda '\). Similarly, let \({\mathcal {S}}_h^\ell \), resp. \({\mathcal {S}}_h^r\), denote the portion of \({\mathcal {S}}_h\) to the left, resp. to the right, of the square \(\Lambda '\).
We now define the irreducible components of a fixed contour \(\gamma \) containing \(\Lambda '\). Consider the portion of \(\gamma \) that intersects \( {\mathcal {S}}_v^t\). This must contain at least one crossing, defined as an open contour connecting the opposite vertical sides of \({\mathcal {S}}_v^t\) that is fully contained in the interior of \({\mathcal {S}}_v^t\). Let \(\gamma ^t\) denote the most internal crossing, i.e. the one that lies closest to the square \(\Lambda '\). We repeat the same construction in the strips \({\mathcal {S}}_h^\ell , {\mathcal {S}}_v^b\) and \(S_h^r\), to define \(\gamma ^\ell , \gamma ^b\) and \(\gamma ^r\) as the most internal crossings. We say that \(\gamma ^u\), \(u\in \{t,\ell ,b,r\}\), form the irreducible components of the contour \(\gamma \). We call \(x^u,y^u\) the endpoints of \(\gamma ^u\), with \(x^u\) coming after \(y^u\) if \(\gamma ^u\) is given a counter clockwise orientation. See Fig. 5. It is easy to convince oneself that any contour containing the square \(\Lambda '\), such that its irreducible components coincide with the given \(\gamma ^t,\gamma ^\ell , \gamma ^b,\gamma ^r\), must have the following property: If we travel along \(\gamma ^t\) in the direction \(y^t \rightarrow x^t\), and then follow the contour, the irreducible components we meet are, in order: \(\gamma ^\ell \) in the direction \(y^\ell \rightarrow x^\ell \), then \(\gamma ^b\) in the direction \(y^b \rightarrow x^b\), then \(\gamma ^r\) in the direction \(y^r \rightarrow x^r\), and finally again \(\gamma ^t\) in the direction \(y^t \rightarrow x^t\). Thus we can write any \(\gamma \) with given irreducible components \(\gamma ^t,\gamma ^\ell , \gamma ^b,\gamma ^r\) as the composition
where \(\eta ^{u,v}\) denotes a path connecting \(x^u\) and \(y^v\) for \(u,v\in \{t,\ell ,b,r\}\).
Let \(\gamma _1,\ldots , \gamma _N\) denote a collection of nested contours as in (5.4). We write \(\gamma _i^u\) for the corresponding irreducible components, and \(\eta _i^{u,v}\) for the remaining components. Clearly, by applying the decomposition (5.6) for each i, one has
where the sum ranges over \(u\in \{t,\ell ,b,r\}\).
Next, we want to decouple the four irreducible pieces, by writing \(\Psi _{\Lambda _M}(\gamma _1,\ldots ,\gamma _N)\) as the sum of a main term \(\sum _u\Psi _u(\gamma ^u_1,\ldots ,\gamma ^u_N)\) and a correction term associated to the corner pieces \(\eta _i\) and to the interactions between distinct irreducible regions. To this end it will be convenient to enlarge the strips \({\mathcal {S}}_v,{\mathcal {S}}_h\) by an amount of order \((\log L)^2\). This will ensure that the expression (5.5) factorizes (up to lower order terms) into the product of four distinct pieces which, see Lemma 5.4 below, can each be reinterpreted as probabilities from the SOS staircase ensemble defined in Sect. 2.5. To define the potential \(\Psi _u(\gamma ^u_1,\ldots ,\gamma ^u_N)\) we proceed as follows.
We start with \(u=t\). Let \({\mathcal {S}}_v'\) denote the infinite vertical strip obtained by enlarging the original strip \({\mathcal {S}}_v\) by \((\log L)^2\):
Let \({\hat{x}}^t_i\) denote the point on the left boundary of \({\mathcal {S}}_v'\) which has the same vertical coordinate as \(x^t_i\) and let \({\hat{y}}^t_i\) denote the point on the right boundary of \({\mathcal {S}}_v'\) which has the same vertical coordinate as \(y^t_i\). Let \({\hat{\gamma }}^t_i\) denote the open contour joining \(\hat{x}^t_i\) and \({\hat{y}}^t_i\) obtained by connecting \({\hat{x}}^t_i\) and \(x^t_i\) by a straight line, then using \(\gamma ^t_i\) from \(x^t_i\) to \(y^t_i\) and then connecting \(y^t_i\) and \({\hat{y}}^t_i\) by a straight line; see Fig. 6. This defines a set of ordered, non-crossing paths \({\hat{\gamma }}^t_i\), \(i=1,\ldots ,N\) in the strip \({\mathcal {S}}_v'\), all staying above the square \(\Lambda '\). For a given choice of \(\gamma ^t_1,\ldots ,\gamma ^t_N\), we define the potential:
where \(L'=\lfloor (1-\delta )L+ (\log L)^2\rfloor \) is half the width of the strip \({\mathcal {S}}_{v}'\), and \(\Phi _{L',\infty }\) is defined in (2.15). The potentials \(\Psi _u(\gamma ^u_1,\ldots ,\gamma ^u_N)\), for \(u=\ell ,b,r\) are defined in the very same way, with the obvious modifications.
Lemma 5.3
Let \(\Psi _{\Lambda _M}\) denote the potential from (5.5). There exists \(\beta _0,C>0\) such that: for any choice of \(\gamma _1,\ldots ,\gamma _N\) in (5.4) with \(\gamma _1\subset \Lambda _{L^2/2}\), for any \(\beta \geqslant \beta _0\) one has
Proof
We are going to use the properties of the potentials listed in Lemma 2.2. In particular, we use the fact that for \(\beta \) large enough, for any \(\Gamma \subset {\mathbb {Z}}^2\), any \(\lambda >0\) one has
for some constant \(C>0\). In the potential \(\Psi _{\Lambda _M}\) one has a sum over subsets \(V\subset \Lambda _M\), while the potential \( \Psi _u\) contains sums over V in the corresponding strips of width \(2L'\). Since we assume \(\gamma _1\subset \Lambda _{L^2/2}\), one has that \(d(\gamma _1,\Lambda _M^c)>L^2/4\) and therefore adding all V’s which are not contained in \(\Lambda _M\) does not change the value of \(\Psi _{\Lambda _M}(\gamma _1,\ldots ,\gamma _N)\) by more than a constant. Similarly, using the fact that there are \(N=O(\log L)\) contours and that \(\gamma ^t_i\) is at distance at least \(\lambda =(\log L)^2\) from the complement of \( {\mathcal {S}}'_v\), when we compute \(\Psi _t\), we may remove the constraint that \(V\subset {\mathcal {S}}'_v\) at the cost of an additive term \(O((\log L)^3)\). Indeed, separating the contribution from the straight pieces in \({\hat{\gamma }}^t_i\), and observing that \(\max _i |\gamma ^t_i|\leqslant CL^2 \) (since all contours belong to \(\Lambda _M\), with \(M=L^2\)) one has that the sum over all \(V\not \subset {\mathcal {S}}'_v\) at distance less than 1 from \(\cup _{i=1}^N{\hat{\gamma }}_i^t\) contributes at most
The same applies to all \(\Psi _u\), \(u\in \{t,\ell ,b,r\}\). The same reasoning shows that the sum over all V’s such that V intersects both \(\gamma ^u_i\) and \(\gamma ^v_j\), for arbitrary i, j is at most a constant if \(u\ne v\). It remains to deal with the contribution from all the V’s which intersect some corner term \(\eta _i^{u,v}\). By the rough bound (5.9) these can be estimated by \(C|\eta _i^{u,v}|\). Putting together these facts one arrives at (5.8). \(\square \)
From (5.5), if \(\gamma _1\subset \Lambda _{L^2/2}\), then Lemma 5.3 implies for \(\beta \) large enough:
Let us now go back to (5.4). Using a very rough bound one can easily obtain
Indeed, write the expansion (2.6) with only one contour and estimate the decoration term \(|\psi _\Lambda (\gamma _1)|\leqslant c_\beta |\gamma _1|\), with a constant \(c_\beta >0\) that vanishes as \(\beta \rightarrow \infty \), and then use a simple Peierls’ argument together with the fact that \(\gamma _1\not \subset \Lambda _{L^2/2}\) implies \(|\gamma _1|\geqslant L^2/2\).
From (5.11) and (5.10), summing over all choices of the points
one has that up to the additive error term \(e^{-L^2}\), \(\widetilde{\mathbb {P}}(A(\delta ,K))\) is upper bounded by
where
and
The sum in (5.13) ranges over all open contours \(\gamma _i^u:y_i^u\rightarrow x_i^u\) such that \(\gamma _{i}^u,\gamma _{j}^u \) do not cross for \(i\ne j\) and such that \(\gamma _{i}^u\) is more internal (closer to \(\Lambda '\)) than \(\gamma _{j}^u\) for \(i>j\). Since we are doing an upper bound, we may neglect the constraint that \(\gamma _i^u\) does not cross the boundary of \(\Lambda '\). The sum in (5.14) ranges over all paths from \(x_i^u\rightarrow y_i^v\). The following lemma summarizes the main estimate we need.
Lemma 5.4
For any u, uniformly in the choice of the points \(x^u,y^u\), one has
Let us conclude the proof by assuming the validity of Lemma 5.4. From (5.14) one has
for some new constant C. Therefore, one has the upper bound
From (5.4)–(5.12), using the uniform bound (5.15) for each u, one has
Since \(N=\frac{1}{4\beta }\log L(1+o(1))\) the conclusion (5.3) follows.
5.2 Proof of Lemma 5.4
The core of the proof is the monotonicity argument of Theorem 4.1 that allows us to consider each of the N contours separately; see Sect. 4. To be able to apply this we first need to reformulate the problem in terms of SOS contours. Without loss of generality we assume that \(u=t\). Let \(\hat{x}_i^t,\ldots ,{\hat{y}}_i^t\) denote the points on the boundary of \({\mathcal {S}}_v'\) as defined before (5.7), and call \(a_{N-i+1}\) the vertical coordinate of \({\hat{x}}_i^t\) and \(b_{N-i+1}\) the vertical coordinate of \({\hat{y}}_i^t\), \(i=1,\ldots ,N\). Let \({\mathcal {Z}}(a_1,\ldots ,a_N;b_1,\ldots ,b_N; L')\), \(L'=(1-\delta )L+(\log L)^2\), denote the partition function of the N contours in the strip \({\mathcal {S}}_v'\) as defined in Lemma 2.3. We claim that
Let us first conclude the proof of Lemma 5.4 assuming the validity of the estimate (5.16). From (5.16) and Theorem 4.1 we can bound \({\mathcal {Z}}_t(x^t,y^t)\) from above by a product of partition functions of a single contour:
The surface tension bound (2.18) then implies the desired estimate (5.15).
To conclude the proof of Lemma 5.4, it remains to prove (5.16). To this end, observe that by the expansion (2.15), one has
where the sum ranges over all collections of non-crossing contours \({\hat{\gamma }}_i:{\hat{x}}_i^t\rightarrow {\hat{y}}_i^t\). Let us restrict this summation to paths of the form \({\hat{\gamma }}_i = {\hat{\gamma }}^t_i\), i.e. paths which have a straight line from \({\hat{x}}_i^t\) to \(x_i^t\), a regular path \(\gamma _i^t:x_i^t\rightarrow y_i^t\), and a straight line from \(y_i^t\rightarrow {\hat{y}}_i^t\); see Fig. 6. By summing over the regular parts \(\gamma _i^t\) and using \(|{\hat{\gamma }}^t_i|=|\gamma ^t_i|+2(\log L)^2\) one has
By the definition (5.7), one has \(\Phi _{L',\infty }({\hat{\gamma }}^t_1,\ldots ,{\hat{\gamma }}^t_N) = \Psi _t(\gamma ^t_1,\ldots ,\gamma ^t_N)\). Therefore, using \(N\leqslant (4\beta )^{-1}\log L\), we conclude
This ends the proof of (5.16).
Notes
After completing this work we realized that a conceptually similar argument was put forward by Bricmont et al. [4, Appendix 1] to compare the step free energy to the free energy associated to a single macroscopic step in the boundary condition.
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Caputo, P., Martinelli, F. & Toninelli, F.L. On the probability of staying above a wall for the \((2+1)\)-dimensional SOS model at low temperature. Probab. Theory Relat. Fields 163, 803–831 (2015). https://doi.org/10.1007/s00440-015-0658-0
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DOI: https://doi.org/10.1007/s00440-015-0658-0
Keywords
- SOS model
- Loop ensembles
- Random surface models
- Entropic repulsion
- Large deviations
Mathematics Subject Classification
- 60K35
- 60F10
- 82B41
- 82C24





