A Peierls’ estimates and low-temperature expansion

Dynamics of -dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion\thanksrefT1

Abstract

We study the Glauber dynamics for the Solid-On-Solid model above a hard wall and below a far away ceiling, on an box of with zero boundary conditions, at large inverse-temperature . It was shown by Bricmont, El Mellouki and Fröhlich [J. Stat. Phys. 42 (1986) 743–798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height . As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height to within an additive constant: . We then show that starting from zero initial conditions the surface rises to its final height through a sequence of metastable transitions between consecutive levels. The time for a transition from height , , to height is roughly for some constant . In particular, the mixing time of the dynamics is exponentially large in , that is, . We also provide the matching upper bound , requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques. Finally, to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in .

[
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10.1214/13-AOP836 \volume42 \issue4 2014 \firstpage1516 \lastpage1589 \fnbelowfloat \newproclaimquatation[theorem]Reader’s Guide \newproclaimremark[theorem]Remark \newproclaimremarks[theorem]Remarks \newproclaimdefinition[theorem]Definition \newproclaimnotation[theorem]Notation

\runtitle

Dynamics of SOS surfaces above a wall

{aug}

A]\fnmsPietro \snmCaputolabel=e1]caputo@mat.uniroma3.it, B]\fnmsEyal \snmLubetzky\correflabel=e2]eyal@microsoft.com, A]\fnmsFabio \snmMartinellilabel=e3]martin@mat.uniroma3.it,C]\fnmsAllan \snmSlylabel=e4]sly@stat.berkeley.edu and D]\fnmsFabio Lucio \snmToninellilabel=e5]toninelli@math.univ-lyon1.fr\thanksreft2 \thankstextT1Supported by the European Research Council through the “Advanced Grant” PTRELSS 228032. \thankstextt2Supported in part by ANR grant SHEPI.

class=AMS] \kwd60K35 \kwd82C20 SOS model \kwdGlauber dynamics \kwdrandom surface models \kwdmixing times

1 Introduction

The -dimensional Solid-On-Solid model is a crystal surface model whose definition goes back to Temperley [47] in 1952 (also known as the Onsager-Temperley sheet). Its configuration space on a finite box with a floor (wall) at , a ceiling at some and zero boundary conditions is the set of all height functions on such that whereas for all . The probability of is given by the Gibbs distribution

(1)

where is the inverse-temperature, denotes a nearest-neighbor bond in the lattice and the normalizing constant is the partition function.

Numerous works have studied the rich random surface phenomena, for example, roughening, localization/delocalization, layering and wetting to name but a few, exhibited by the SOS model and some of its many variants. These include the discrete Gaussian (replacing by for the integer analogue of the Gaussian free field), restricted SOS (nearest neighbor gradients restricted to ), body centered SOS [50], etc. (for more on these flavors see, e.g., [52, 3, 5]).

Of special importance is SOS with , the only dimension featuring a roughening transition. Consider the SOS model without constraining walls (the height function takes values in ). For , it is well known [47, 48, 22] that the SOS surface is rough (delocalized) for any , that is, the expected height at the origin (in absolute value) diverges in the thermodynamic limit . However, for a Peierls argument shows that the surface is rigid (localized) for any (see [10]), that is, is uniformly bounded in expectation. This is also the case for and large enough [8, 28]. That the surface is rough for at high temperatures was established in seminal works of Fröhlich and Spencer [25, 26, 27]. Numerical estimates for the critical inverse-temperature where the roughening transition takes place suggest that .

One of the main motivations for studying an SOS surface constrained between two walls, both its statics and its dynamics, stems from its correspondence with the Ising model in the phase coexistence region. For concreteness, take a box of side-length in with minus boundary conditions on the bottom face and plus elsewhere. One can view the SOS surface taking values in as the interface of the minus component incident to the bottom face, in which case the Hamiltonian in (1) agrees with that of Ising up to bubbles in the bulk. At low enough temperatures bubbles and interface overhangs are microscopic, thus SOS should give a qualitatively correct approximation of Ising (see [2, 22, 41]). Indeed, in line with the SOS picture, it is known [49] that the  Ising model undergoes a roughening transition at some satisfying [where is the critical point for Ising on ], yet there is still no rigorous proof that (see [52] for more details).

When the SOS surface is constrained to stay above a hard wall (or floor), Bricmont, El Mellouki and Fröhlich [9] showed in 1986 the appearance of the entropic repulsion: for large enough , the floor pushes the SOS surface to diverge even though . More precisely, using Pirogov–Sinaï theory (see the review [45]), the authors of [9] showed that the SOS surface on an box rises, amid the penalizing zero boundary, to an average height satisfying for some absolute constant , in favor of freedom to create spikes downwards.

Entropic repulsion is one of the key features of the physics of random surfaces. This phenomenon has been rigorously analyzed mainly for some continuous-height variants of the SOS model in which the interaction potential is replaced by a convex potential ; see, for example, [7, 6, 15, 4, 53, 51], see also [1] for a recent analysis of the wetting transition in the SOS model. As we will see below, entropic repulsion has a profound impact not only on the equilibrium shape of the surface but also on its time evolution under natural Markovian dynamics for the interface. The rigorous analysis of these dynamical effects of entropic repulsion will be the central focus of this work.

The dynamics we consider is the heat bath dynamics, or Gibbs sampler, for the equilibrium measure , that is, the discrete time Markov chain where at each step a site is picked at random and the height of the surface at is replaced by a random variable distributed according to the conditional probability . This defines a Markov chain with state space , reversible with respect to , commonly referred to as the Glauber dynamics. As explained below, our results apply equally well to other standard choices of reversible Markov chains, such as, for example, the Metropolis chain where only moves of the type are allowed.

The mixing time is defined as the number of steps needed to reach approximate stationarity with respect to total variation distance, see Section 2 for definitions.

The main result of this paper is that the mixing of Glauber dynamics for the SOS is exponentially slow, due to the nature of the entropic repulsion effect.

Theorem 1

For any sufficiently large inverse-temperature there is some such that the following holds for all . The mixing time of the Glauber dynamics of the SOS model on with zero boundary conditions, floor at zero and ceiling at with satisfies

(2)

The exponentially large mixing time in (2) is in striking contrast with the rapid mixing displayed by Glauber dynamics of the SOS model [38, 12]. When it is known that the main driving effect is a mean-curvature motion which induces a diffusive relaxation to equilibrium, with of order up to ) corrections. As we will see, in instead the main mechanism behind equilibration is a series of metastable transitions through an increasing series of effective energy barriers caused by the entropic repulsion. This is also in contrast with the behavior of related interface models with continuous heights as, for example, in [16, 20].

1.1 Metastability and entropic repulsion

Consider the evolution of an initially flat surface at height zero. We shall give a rough description of how it rises to the final height through a series of metastable states indexed by . Roughly speaking the surface in state with label is approximately flat at height with rare up or downward spikes. Of course downward spikes cannot be longer than because of the hard wall. If then the surface has an advantage to rise to the next level . This is due to the gain in entropy, measured by the possibility of having downward spikes of length , beating the energy loss from the zero boundary conditions.

The mechanism for jumping to the next level should then be very similar to that occurring in the Ising model at low temperature with a small external field opposite to the boundary conditions (see [43, 44]). Specifically, via a large deviation the surface at height creates a large enough droplet of sites at height which afterwards expands to cover most of the available area. The energy/entropy balance of any such droplet is roughly1 of order where and are the boundary length and area, respectively, and the effective field represents the probability of a isolated downward spike. Simple considerations suggest then that the critical length of a droplet should be proportional to . Finally, the well-established metastability theory for the Ising model indicates that the activation time for such a critical droplet should be exponential in the critical length2 (i.e., a double exponential in ) as seen in Figure 1.

Of course, in order to establish, even partially, the above picture and to prove the asymptotic of as per (2) it is imperative to estimate the final equilibrium height of the surface to within an additive . In Section 3 (Theorem 3.1), we improve the estimates of [9] to show that in fact the typical height of the surface at equilibrium is , where

(3)

The aforementioned picture of the evolution of the SOS surface through a series of metastable states is quantified by the following result.

Figure 1: Illustration of the series of metastable states in the surface evolution. The dynamics waits time until the formation of a macroscopic droplet (marked in red) which eventually raises the average height from to .
Theorem 2

For any sufficiently large inverse-temperature there is some such that the following holds. Let be the Glauber dynamics for the SOS model on with zero boundary conditions, floor at zero and ceiling at with , started from the all-zero initial state. Fix and let where

(4)

Then and yet

(5)

In fact, we prove this with the constant in (4) replaced by where . Moreover, the statement of the above theorem remains valid when as long as the target level satisfies for some sufficiently large .

{remark*}

A natural conjecture in light of Theorem 2 is that there exists a constant such that the distribution of converges as to an exponential random variable.

We wish to emphasize that, as will emerge from the proof, the exponential slowdown of equilibration is a coupled effect of entropic repulsion and of the rigidity of the interface. In particular, the following rough upper bound shows that the situation is very much different when the floor constraint is absent (yet the ceiling constraint remains unchanged).

Theorem 3

Consider the SOS setting as in Theorem 1 with the exception that the surface heights belong to the symmetric interval. Then .

Specifically, our proof gives the estimate . No effort was made to improve the exponent as we would expect the true mixing behavior to be polynomial in . We further expect that in the presence of a floor yet for the mixing time will have a different scaling with the side-length .

It is useful to compare our results with those of [13], where the Glauber dynamics for the SOS above a hard wall, at low temperature and in the presence of a weak attracting (towards the wall) external field was analyzed in details. There it was proved that certain critical values of the external field induce exponentially slow mixing while for all other values the dynamics is rapidly mixing. Although the slow mixing proved in [13] is similar to the one appearing in (2), the physical phenomenon behind it is very different. When an external field is present, a critical value of it results in two possible and roughly equally likely heights for the surface. In this case, slow mixing arises because of the presence of a typical bottleneck in the phase space related to the bi-modal structure of the equilibrium distribution. In the setting of Theorems 1 and 2 instead, there is in general no bi-modal structure of the Gibbs measure and the slow mixing takes place because of a multi-valley structure of the effective energy landscape induced by the entropic repulsion which produces a whole family of bottlenecks.

1.2 Methods

We turn to a description of the main techniques involved in the proof of the main theorems. Our results can be naturally divided into three families: equilibrium estimates, lower bounds on equilibration times, and upper bounds on equilibration times.

Equilibrium estimates

Our proof begins by deriving estimates for the equilibrium distribution which are crucial to the understanding of the dynamics (as discussed in Section 1.1) and of independent interest. Over most of the surface, the height is concentrated around as defined in (3) with typical fluctuations of constant size. Achieving estimates with a precision level of an additive turns out to be essential for establishing the order of the mixing time exponent: indeed, analogous estimates up to some additive tending to with would set off this exponent by a factor of .

The main techniques deployed for this part are a range of Peierls-type estimates for what we refer to as -contours, defined as closed dual circuits with values at least on the sites along their interior boundary and at most along their exterior boundary. In the simpler setting of no floor or ceiling (i.e., the sites are free to take all values in as their heights), the map which decreases all sites inside an -contour by 1 is bijective and increases the Hamiltonian by , the length of the contour. Hence, the probability of a given -contour in this setting is bounded by . Iterating estimates of this form allows us to bound the deviations of the sites with the correct asymptotic in the setup of having no walls.

The presence of a floor renders this basic Peierls argument invalid since the map may leave sites in the interior with negative values. Rather than a technicality, this in fact lies at the heart of the entropic repulsion effect. We resort to estimating the probability that a given -contour has a strictly positive interior, a quantity directly involving its area. By analyzing an isoperimetric tradeoff between the contour’s area and perimeter, we show that large contours above height are unlikely, which in turn implies typical fluctuations above this level. For a lower bound on the typical height of the surface we show that if too many sites are below then the loss in energy due to raising the entire surface by 1 is more than compensated by the increased entropy from the freedom to create downward spikes reaching 0. Put together, these estimates guarantee that the height of most sites is within a constant of .

Equilibration times: Lower bounds

Fix with and consider the restricted ensemble obtained by conditioning the equilibrium measure on the event that all -contours have area smaller than , for some small . Our equilibrium estimates imply that in this restricted ensemble: {longlist}[(iii)] (i) each -contour is actually very small [e.g., with area less than ], with very high probability; (ii) the probability of the boundary of is ; (iii) the probability of having a large density of heights at least is . In some sense (i), (ii) and (iii) above establish a bottleneck the Markov chain must pass through and thus provide the sought lower bound of on the typical value of the hitting time in Theorem 2 when the initial state is the all zero configuration. In fact, the initially flat configuration can be replaced by monotonicity by the restricted ensemble described above. Then, in order for to be smaller than , either the dynamics has gone through the boundary of before or the event described in (iii) occurred without leaving . Either way an event with probability occurred and the minimal time to see it must be proportional to the inverse of its probability.

Equilibration times: Upper bounds

By the monotonicity of the system, it is enough to consider the chain starting from the maximum and minimum configurations. The natural approach is to apply the well-known canonical paths method (see [17, 18, 30, 46] for various flavors of the method). As the cut-width of the cube is , the most naïve application of this approach would give a bound of . A better bound can be shown by considering the problem with maximum height . In this case, the cut-width is of order yielding a mixing time upper bound of . Since the height fluctuations are logarithmic, we can iterate this analysis using monotonicity and censoring to get a bound of for the original model with , vs. our lower bound of . However, removing the factor that separates these exponents entails a significant amount of extra work.

The basic structure of the proof is to first establish a burn-in phase where we show that, starting from the maximal and minimal configurations, the process reaches a “good” set featuring small deviations from the equilibrium level . From there, we establish a modified canonical paths estimate (Theorem 2.3), showing that it is enough to establish a reasonable probability of hitting the good set from any starting location together with a good canonical paths estimate restricted to this set. This new tool, which we believe is of interest on its own right, is described in detail in Section 2.3 and proved in a general context in Section 5.

Showing that the surface falls down from the ceiling (the maximum height) to , as depicted in Figure 2, ought to have been the easier part of the burn-in argument since high above the floor there is no entropic repulsion effect. Unfortunately a number of major technical challenges must be overcome.

Figure 2: Glauber dynamics for SOS on a square lattice at from an initial state . Surface gradually falls towards level . Snapshots at (top left), , , 10,000 (bottom right) in cont. time.

First, the effect of the entropic repulsion is still apparent for the estimates we require when the surface is fairly close to . To overcome this, we add a small external field to the model, thereby modifying the mixing time by a factor of at most (which is large but still of the same order as our designated upper bound) and tilting the measure to remove these entropic repulsion effects. Second, while our main equilibrium estimates were proved using Peierls-type estimates, for the burn-in we require some of the cluster expansion machinery of [19] which we extend to the SOS framework. This involves a number of challenges including showing that the contours we consider do not interact significantly with the boundary conditions, a highly nontrivial fact. Implementing this scheme is the biggest challenge of the paper and we provide extensive notes for the reader in these sections to explain the rather technical proofs.

Finally, the fact that the surface rises from the floor (the all zero initial condition) to the vicinity of the equilibrium height in time is proved via an unusual inductive scheme. Unlike other multi-scale inductive schemes, somewhat surprisingly the one used here does not incur any penalizing factor on the upper bound. We first prove weaker bounds on the mixing time and use these estimates to show that a smaller box of side-length mixes by time . By monotonicity, we can use this to bound the distance from the equilibrium height of the surface in the original box by . By using this height estimate along with our canonical paths result, we get improved bounds on the mixing time. This in turn allows us to take larger sub-boxes and iteratively achieve better and better estimates on the distance to . After sufficiently many iterations, we show that the surface reaches height in time and thereafter the canonical paths estimate completes the proof.

1.3 Related open problems

Tilted walls

An interesting and to our knowledge widely open problem concerns the SOS model with a nonhorizontal hard wall, that is, when the constraint is replaced by , where denotes the discrete approximation of the plane orthogonal to the unit vector , and is assumed to have all components different from zero. The equilibrium fluctuations for can be analyzed via their representation through dimer coverings [31] and the variance of the surface height in the middle of the box can be shown to be ; see [11], Section 5, for a proof. Moreover, at , as far as the dynamics is concerned, it has been proved [12] that the mixing time is of order up to corrections and that the relaxation process is driven by mean curvature motion. The case , however, remains open both for equilibrium fluctuations and for mixing time bounds.

Mixing time for Ising model

In view of the natural connection with the Ising model, the study of Glauber dynamics for the SOS can also shed some light on a, still open, central problem in the theory of stochastic Ising models: its mixing time under an all-plus boundary in the phase coexistence region. The long-standing conjecture is that the mixing time of Glauber dynamics for the Ising model on a box of side-length with all-plus boundary should be at most polynomial in at any temperature. More precisely, the convergence to equilibrium should be driven by a mean-curvature motion of the interface of the minus droplet in accordance with Lifshitz’s law [35]. For instance, the mixing time of Glauber dynamics for Ising on an square lattice is conjectured [21] to be of order in continuous time. This was confirmed at zero temperature [14, 23, 33] and near-zero temperatures [11], yet the best-known upper bound for finite remains quite far, a quasi-polynomial bound of due to [36]. The understanding of Ising is far more limited: while at zero temperature bounds of were recently proven in [11], no sub-exponential mixing bounds are known at any finite .

2 Definitions and tools

2.1 Glauber dynamics for solid-on-solid

Let and denote the minimal and maximal configurations in , that is, and for every . Given a finite connected subset , let denote its external boundary, that is, the set of sites in which are at distance from . To extend the SOS definition to arbitrary boundary conditions (b.c.) given by , define the SOS Hamiltonian with b.c. to be

(6)

Given and , the Gibbs measure on with b.c. is defined as

(7)
{notation}

In the sequel when the b.c. we will use the abbreviated form . We will occasionally drop the subscript and superscript from the notation of when there is no risk of confusion. Moreover, we will need to address the following variants of : {longlist}[(iii)] (i) the measure of SOS without walls (no floor and no ceiling) and with b.c. at height ; (ii) the measure corresponding to with (no ceiling); (iii) starting from Section 6 the measures (and its analog with no ceiling) corresponding to the SOS Hamiltonian with an additional external field of the form with for some fixed constant [see, e.g., (50)]. The dynamics under consideration is a discrete-time Markov chain, defined as follows. To construct given , pick a site uniformly at random; sample a new value for from the equilibrium measure conditioned on the current heights at the neighboring sites, that is, . The law of the process with initial condition is denoted by , the configuration at time is and its law is . When there is no need to emphasize the initial condition, we simply write for the configuration at time . It is well known that this Markov chain is reversible w.r.t. the invariant measure . The mixing time is defined to be the time the process takes to converge to equilibrium in total variation distance, that is, (8) where denotes the total variation distance between two measures . It is well known (e.g., [34], Section 4.5) that the total variation distance from equilibrium decays exponentially with rate , namely (9) The relaxation time is the inverse of the spectral gap of the transition kernel of the chain. The spectral gap, denoted by , has the following variational characterization: (10) where is the transition kernel of the chain, is the identity matrix and the infimum is over all nonconstant functions . The following standard inequality (see, e.g., [34], Section 12.2, and [42]) relates the mixing time and the relaxation time: (11) with . By definition, in the SOS model and , thus for large enough (12)

From now on we refer to the Markov chain defined above as the Glauber dynamics. One can use standard comparison estimates to obtain equivalent versions of our main results for other standard choices of Markov chains that are reversible w.r.t. the SOS Gibbs measures, such as, for example, the Metropolis chain with 1 updates. Indeed, since the heights are confined within an interval of size it is not hard to see that the ratio between the different mixing times is at most polynomial in . We refer to, for example, [11], Section 6, for a detailed argument in this direction.

2.2 Monotonicity

Our dynamics is monotone (or attractive) in the following sense. One equips the configuration space with the natural partial order such that if for every . It is possible to couple on the same probability space the evolutions corresponding to every possible initial condition and boundary condition in such a way that if and then for every . Here, we indicated explicitly the dependence on the boundary conditions but we will not do so in the following. The law of the global monotone coupling is denoted .

A first consequence of monotonicity is that the FKG inequalities [24] hold: if and are two increasing (w.r.t. the above partial ordering) functions, then and the same holds for the measure without the floor/ceiling.

Monotonicity also implies the following standard fact [cf., e.g., the proof of [39], equation (2.10)]: for every initial condition and boundary condition ,

(13)

Another consequence of monotonicity is the so-called Peres–Winkler censoring inequality. Take integers , a sequence of and . Consider the following modified dynamics . To construct given ,

  • pick a site uniformly at random;

  • at time with do as follows:

    • if or if and then do nothing;

    • if and then replace its value with a new value in with probability proportional to the stationary measure conditioned on the value of the neighboring columns,

Call the law at time when the initial distribution is . The following then holds:

Theorem 2.1 ((Special case of [40], Theorem 1.1))

If the initial distribution is such that is an increasing (resp., decreasing) function, then is also increasing (resp., decreasing) for and (resp., ). In addition,

(14)

2.3 An improved path argument

Geometric techniques can prove very effective in getting upper bounds on the relaxation time and therefore on the mixing time of a Markov chain [17, 18, 30, 46] (see also [34], Section 13.5). Let us recall the basic principle.

Let be a discrete-time reversible Markov chain on a finite state space , with invariant measure . For such that the one-step transition probability from to is nonzero, set . For each couple , fix a path in with , and and let . Then the relaxation time of the Markov chain is bounded as

(15)

Here, means that if then there exists  such that , . The proof is simply an application of the Cauchy–Schwarz inequality; see, for example, [42].

An application of this principle gives the following proposition.

Proposition 2.2

For the SOS dynamics in the , , with floor at height zero, ceiling at and b.c. , one has for some

(16)

and, thanks to (12), if .

That (16) easily follows from (15) was observed in [37] in the case of the Glauber dynamics of the Ising model (in this case one refers to the paths as “canonical paths”). For SOS the proof is very similar and is given for completeness in Section 5.1.

However, this upper bound is too rough for our purposes since we have while we wish to get a mixing time upper bound which is exponential in . Therefore, a significant part of the present work is devoted to getting rid of the nonphysical factor in the argument of the exponential in the r.h.s. of (16). Although this task may appear to be mainly of technical nature it actually requires a much deeper understanding of the actual behavior of the dynamics compared to that provided by canonical paths, and the support of new ideas.

One of the key ingredients we use is the following improved version of (15), which we believe can be interesting in a more general context.

Theorem 2.3

Let and assume that, for some and for every initial condition , with denoting the law of the chain starting at . Assume further that for every in there exists a path as above which stays in and let

(17)

Then,

(18)

with .

This is clearly an improvement provided that is bounded away from zero, that and that is not too large (in simple words, we need that with nonzero probability the chain enters “quickly” the good set where canonical paths work well).

In our SOS application, roughly speaking, we will choose to be the set of configurations such that is upper bounded by a constant. We will see that, irrespective of the starting configuration, at time the dynamics is in with probability at least . On the other hand, a minor modification of Proposition 2.2 will give . Then, Theorem 2.3 allows us to improve the mixing time upper bound to .

3 Equilibrium results

Theorem 3.1

Let be a box of side-length and let . Set . There exist some absolute constants (with integer) such that for any integer ,

(19)
(20)

(Notice that the bound on downward fluctuations improves with the size of the deviation whereas the bound on upward fluctuations deteriorates with the distance.)

Recall that has a floor at and a ceiling at height (together with zero boundary conditions). It will be convenient throughout this section to work in the setting of a floor at 0 but no ceiling, where the corresponding measure is asymptotically equal to .

Lemma 3.2

There is an absolute constant such that for any and any subset of configurations ,

The above lemma, which will be proved further on in this section, entitles us to derive results on from at an asymptotically negligible cost.

The following notion of a contour and that of an -contour, a level line at height , play a crucial role in our proofs.

{definition}

We let be the dual lattice of and we call a bond any segment joining two neighboring sites in . Two sites in are said to be separated by a bond if their distance (in ) from is . A pair of orthogonal bonds which meet in a site is said to be a linked pair of bonds if both bonds are on the same side of the forty-five degrees line across . A geometric contour (for short a contour in the sequel) is a sequence of bonds such that:

    [(3)]
  1. for , except for and where ;

  2. for every , and have a common vertex in ;

  3. if , , , intersect at some , then and are linked pairs of bonds.

We denote the length of a contour by , its interior (the sites in it surrounds) by and its interior area (the number of such sites) by . Moreover, we let be the set of sites in such that either their distance (in ) from is , or their distance from the set of vertices in where two nonlinked bonds of meet equals . Finally, we let and .

{definition}

Given a contour we say that is an -contour for the configuration if

We will say that is a contour for the configuration if there exists such that  is a -contour for . Finally, will denote the event that is an -contour.

To illustrate the above definitions with a simple example, consider the elementary contour given by the square of side surrounding a site . In this case, is an -contour iff and for all . In general, (resp., ) is the set of (resp., ) either at distance 1 from (resp., ) or at distance from a vertex (resp., ) in the south–west or north–east direction.

{remark}

As the reader may have noticed the definition of an -contour is asymmetric in the sense that we require the minimal height of the surface at the inner boundary of , , to be larger than the maximum height at the external boundary. In a sense, this definition covers upward fluctuations of the surface. Of course one could provide the reverse definition covering downward fluctuations. In the sequel, the latter is not really needed thanks to monotonicity and symmetry arguments. We also observe that, contrary to what happens in, for example, Ising models, a geometric contour could be at the same time a -contour and a -contour with . More generally two geometric contours could be contours for the same surface with different height parameters even if (but one of them must be contained in the other).

The following estimates play a key role in the proof of Theorem 3.1.

Proposition 3.3

There exists an absolute constant such that for all and ,

(21)

Moreover, for any family of -contours such that for all

and when , ,we have

(22)

As a step towards the proof of the above proposition, we consider the setting of no floor and no ceiling, where the picture is simpler as there is no entropic repulsion.

Lemma 3.4

For any -contour in any domain with any boundary condition we have

Moreover, if and are contours with then

(23)
{pf}

Define the map by

(24)

If has an -contour at , then the difference along every edge in crossing decreases by 1 so . Since is a bijection it follows that

Equation (23) follows from the same argument by noting that if then remains in . This completes the proof.

{remark}

In the context of considering the interior of an -contour for possibly nested contours [such as the ones featured in equation (23)], a useful observation is that

for any boundary condition , that is, at most all along . This follows from the fact that conditioning on any fixed would contribute an equal pre-factor to all configurations thanks to having , and as this includes all ’s with this further includes . Moreover, the same holds when conditioning on (instead of just ) for an arbitrary event which is only a function of the configuration on .

(Note that the above remark similarly applies to and by the same argument.)

A Peierls-type argument will transform the above lemma into the following bound on upward (downward) fluctuations in the no floor, no ceiling setting.

Proposition 3.5

There exists an absolute constant such that for any , domain , site and height ,

{pf}

Define the map by for and

Observe that and that since changes the Hamiltonian by at most ,

Moreover, as is injective, summing over we have that

Since by symmetry the lower bound follows.

To get the upper bound, define a set of nested contours surrounding as

and observe that, if is such that , then necessarily there exists such that .

Applying Lemma 3.4 iteratively (while bearing Remark 3 in mind), we now obtain that for every ,

(25)

Simple counting gives that the number of contours of length starting from a vertex is at most , the number of self avoiding walks of length . If such a path surrounds , then it must cross the horizontal line containing to its right within distance so the number of with and is at most (with room to spare). Hence

which is uniformly bounded in for any since the connective constant is known to satisfy . Hence, for some large enough , independent of ,

(26)

Now define a collection of nested contours of area at least 2 and at most as

We note that