Quasipolynomial mixing of the 2d stochastic Ising model with “plus” boundary up to criticality
Abstract.
We considerably improve upon the recent result of [cf:MT] on the mixing time of Glauber dynamics for the 2d Ising model in a box of side at low temperature and with random boundary conditions whose distribution stochastically dominates the extremal plus phase. An important special case is when is concentrated on the homogeneous allplus configuration, where the mixing time is conjectured to be polynomial in . In [cf:MT] it was shown that for a large enough inversetemperature and any there exists such that . In particular, for the allplus boundary conditions and large enough .
Here we show that the same conclusions hold for all larger than the critical value and with replaced by (i.e. quasipolynomial mixing). The key point is a modification of the inductive scheme of [cf:MT] together with refined equilibrium estimates that hold up to criticality, obtained via duality and randomline representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which quantitatively sharpen the Brownian bridge picture established e.g. in [cf:GI, cf:Higuchi, cf:Hryniv].
Key words and phrases:
Ising model, Mixing time, Phase coexistence, Glauber dynamics.2010 Mathematics Subject Classification:
60K35, 82C201. Introduction
The Ising model on lattices at and near criticality has been the focus of numerous research papers since its introduction in 1925, establishing it as one of the most studied models in mathematical physics. In two dimensions the model was exactly solved by Onsager [cf:Onsager] in 1944, determining its critical inversetemperature in the absence of an external magnetic field. While the classical study of the Ising model concentrated on its static properties, over the last three decades significant efforts were dedicated to the analysis of stochastic dynamical systems that both model its evolution and provide efficient methods of sampling it. Of particular interest is the interplay between the behaviors of the static and dynamical models as they both undergo a phase transition at the critical .
The Glauber dynamics for the Ising model (also known as the stochastic Ising model), introduced by Glauber [cf:Glauber] in 1963, is considered to be the most natural sampling method for it, with notable examples including heatbath and Metropolis. It is known that on a box of sidelength in with free boundary conditions (b.c.), alongside the phase transition in the range of spinspin correlations in the static Ising model around , the corresponding Glauber dynamics exhibits a critical slowdown: Its mixing time (formally defined in §1.1) transitions from being logarithmic in in the high temperature regime to being exponentially large in in the low temperature regime , en route following a power law at the critical .
One of the most fundamental open problems in the study of the stochastic Ising model is understanding the system’s behavior in the socalled phasecoexistence region under homogenous boundary conditions, e.g. allplus boundary. In the presence of these b.c. the phase becomes unstable and as such the reduced bottleneck between the two phases drastically accelerates the rate of convergence of the dynamics to equilibrium. Indeed, in this case the Glauber dynamics is known to mix in time that is subexponential in the surface area of the box, contrary to its lowtemperature behavior with free boundary. The central and longstanding conjecture addressing this phenomenon states that the mixing time of Glauber dynamics for the Ising model on a box of sidelength with allplus boundary conditions is at most polynomial in at any temperature.
So far this has been confirmed on the 2d lattice throughout the onephase region (see [cf:MO1, cf:MO2]) and very recently at the critical (see [cf:LS]). Despite intensive efforts over the last two decades, establishing a powerlaw behavior for the mixing of Glauber dynamics at the phasecoexistence region under the allplus b.c. remains an enticing open problem.
In [cf:FH] the precise order of mixing in this regime on a 2d square lattice of sidelength was conjectured to be in accordance with Lifshitz’s law (see [cf:Lifshitz] and also [cf:CSS, cf:Ogielski, cf:Spohn]). The heuristic behind this prediction argues that when a droplet of the phase is surrounded by the phase at low temperature it proceeds to shrink according to the meancurvature of the interface between them. Unfortunately, rigorous analysis is still quite far from establishing the expected Lifshitz behavior of mixing.
Until recently the best upper bound on the mixing at the phasecoexistence region under the allplus boundary was due to [cf:Martinelli] and valid for large enough . This bound from 1994 was substantially improved in a recent breakthrough paper [cf:MT], where it was shown (as a special case of a result on a wider class of b.c.) that for a sufficiently large and any the mixing time is . The approach of [cf:MT] hinged on a novel inductive scheme on boxes with random boundary conditions, combined with a careful use of the socalled PeresWinkler censoring inequality; these ideas form the foundation of the present paper. Note that the requirement of large in [cf:Martinelli, cf:MT] was essential in order to make use of results of [cf:DKS] on the Wulff construction, available only at low enough temperature by cluster expansion methods. For smaller values of the best known estimates on the mixing time are due to [cf:CGMS] and of the weaker form .
In this work we improve these estimates into an upper bound of the form on the mixingtime (i.e. quasipolynomial in the sidelength ) valid for any . The key to our analysis is a modification of the recursive framework introduced in [cf:MT] combined with refined equilibrium estimates that hold up to criticality. To establish these, in lieu of relying on clusterexpansions, we utilize duality and the randomline representation machinery for the high temperature Ising model developed in [cf:PV1, cf:PV2].
A key new element of our proof concerns fine estimates on the fluctuations of cluster boundaries. Whenever the boundary is pinned at two vertices and , the contour of the cluster is known to converge to the Brownian bridge (cf. [cf:DH, cf:Higuchi, cf:Hryniv]). This does not, however, allow us to estimate the probability of events when these converge to 0 in the limit. In particular, we are interested in: (i) a Gaussian bound for the probability that the contour would reach height (established in Theorem 5.3); (ii) the probability that the contour remains in the upper halfplane, an event that would have probability were the contour to behave like a 1d random walk of length conditioned to return to 0. In §5 (see Theorem 5.1) we prove that up to multiplicative constants this indeed holds for a given contour.
These then provide important tools in estimating the probability of various other events characterizing the Ising interfaces at equilibrium.
1.1. Glauber dynamics for the Ising model
Let be a generic finite subset of . Write for the nearestneighbor relation in (i.e. if ) and define , the boundary of , to be the nearestneighbors of in :
The classical Ising model on with no external magnetic field is a spinsystem whose set of possible configurations is . Each configuration corresponds to an assignment of plus/minus spins to the sites in and has a statistical weight determined by the Hamiltonian
where forms the boundary conditions (b.c.) of the system. The Gibbs measure associated to the spinsystem with boundary conditions is
(1.1) 
where is the inverse of the temperature (i.e. ) and the partitionfunction is a normalizing constant. When the boundary conditions are uniformly equal to (resp. ) we will denote the Gibbs measure by (resp. ). Throughout the paper we will omit the superscript and the subscript from the notation of the Gibbs measure when these are made clear from the context.
The Gibbs measure enjoys a useful monotonicity property that will play a key role in our analysis. Consider the usual partial order on whereby if for all . A function is monotone increasing (decreasing) if implies (). An event is increasing (decreasing) if its characteristic function is increasing (decreasing). Given two probability measures on we say that is stochastically dominated by , denoted by , if for all increasing functions (here and in what follows stands for ). According to these notations the wellknown FKG inequalities [cf:FKG] state that

If then .

If and are increasing then .
The phase transition regime in the 2d Ising model occurs at low temperature and it is characterized by spontaneous magnetization in the thermodynamic limit. There is a critical value such that for all ,
(1.2) 
Furthermore, in the thermodynamic limit the measures and converge (weakly) to two distinct Gibbs measures and which are measures on the space , each representing a pure state. We will focus on this phasecoexistence region .
The Glauber dynamics for the Ising model is a family of continuoustime Markov chains on the state space , reversible with respect to the Gibbs distribution . An important and natural example of this stochastic dynamics is the heatbath dynamics, which we will now describe, postponing the formulation of the general Glauber dynamics to §2.1. Note that our results apply to all of these chains (e.g., Metropolis etc.) by standard arguments for comparing their mixing times (see e.g. [cf:Martinelli97]).
The heatbath dynamics for the Ising model is defined as follows. With a rate one independent Poisson process for each vertex , the spin is refreshed by sampling a new value from the set according to the conditional Gibbs measure
It is easy to verify that the heatbath chain is indeed reversible with respect to and is characterized by the generator
where is the average of with respect to the conditional Gibbs measure acting only on the variable . The Dirichlet form associated to takes the form
where denotes the variance with respect to . It is possible to extend the above definition of the generator directly to the whole lattice and get a well defined Markov process on (see e.g. [cf:Liggett]). The latter will be referred to as the infinite volume Glauber dynamics, with generator denoted by .
We will denote by the distribution of the chain at time when the starting configuration is identically equal to . For instance, for any and the expectation of w.r.t. is given by where is the Markov semigroup generated by . The notation will stand for the corresponding quantity for an initial configuration of either allplus or allminus.
A key quantity that measures the rate of convergence of Glauber dynamics to stationarity is the gap in the spectrum of its generator, denoted by . The Dirichlet form associated with produces the following characterization for the spectralgap:
where the infimum is over all nonconstant . Another useful measure for the speed of relaxation to equilibrium is the totalvariation mixing time which is defined as follows. Recall that the totalvariation distance between two measures on a finite probability space is defined as
For any , the mixingtime of the Glauber dynamics is given by
When we will simply write . This particular definition yields the following wellknown inequalities (see e.g. [cf:SaloffCoste, cf:LPW]):
The last inequality shows that in our setting and are always within a factor of from one another (to see this, observe that for any by Eq. (1.1) whereas ). One could restate our results as well as the analogous conjecture on the polynomial mixing time under allplus b.c. in terms of (expected to have order , the sidelength of , for any ; see [cf:BM, cf:CMST]).
1.2. Main results
We are now in a position to formalize the main contribution of this paper. The following theorem is the counterpart of the main result obtained by two of the authors in [cf:MT]. Here we feature an improved estimate that in addition holds not only for large enough but throughout the phasecoexistence region.
Theorem 1.
For any there exists some so that the following holds for the Glauber dynamics for the Ising model on the square at inversetemperature . If is of the form for some integer then:

If the boundary conditions are sampled from a law that either stochastically dominates the pure phase or is stochastically dominated by then
(1.3) In particular,
(1.4)
The most natural consequence of the above result is obtained when concentrates on homogenous boundary conditions, where the best previous bounds were for any and large enough ([cf:MT]) along with for all other ([cf:CGMS]).
Corollary 2.
For any there exists some so that the mixing time of Glauber dynamics for the Ising model on the square with b.c. satisfies
(1.5) 
The same bound holds if the boundary conditions are on three sides and on the remaining one, and similarly if is replaced by .
We believe that improving the above bound into the conjectured polynomial one would require substantial new ideas. Indeed, in the present recursive framework in which the final scale of the system is reached via a doubling sequence, at each step the mixingtime estimate worsens by a power of (hence the quasipolynomial bound). For a polynomial upper bound one could not afford to lose more than a constant factor on average along these steps.
One may also apply Theorem 1 to deduce the mixing behavior of the 2d Ising model under Bernoulli boundary conditions, as illustrated by the next corollary. Here and in what follows we say that an event holds with high probability (w.h.p.) to denote that its probability tends to as the size of the system tends to .
Corollary 3.
Let and consider Glauber dynamics for the Ising model on the square with b.c. comprised of i.i.d. Bernoulli variables, for some . Then w.h.p. for some .
To obtain the above corollary observe that the Bernoulli boundary conditions with the above specified clearly stochastically dominate the marginal of on .
The mixing time of Glauber dynamics for Ising on a finite box under allplus b.c. is closely related to the asymptotic decay of the time autocorrelation function in the infinitevolume dynamics on started at the plus phase. Here it was conjectured in [cf:FH] that the decay should follow a stretched exponential of the form . As a byproduct of Corollary 2 (and standard monotonicity arguments) we obtain a new bound on this quantity, improving on the previous estimate due to [cf:MT] of with arbitrarily large which was applicable for large enough .
Corollary 4.
Let , let and define to be the time autocorrelation of the spin at the origin started from the plus phase (the variance is w.r.t. the plus phase ). Then there exists some such that for any ,
(1.6) 
1.3. Related work
Over the last two decades considerable effort was devoted to the formidable problem of establishing polynomial mixing for the stochastic Ising model on a finite lattice with allplus b.c. Following is a partial account of related results.
Analogous to its conjectured behavior on , the mixing of Glauber dynamics for the Ising model on the lattice in any fixed dimension is believed to be polynomial in the sidelength of the box at any temperature in the presence of an allplus boundary. Unfortunately, the stateoftheart rigorous analysis of the problem in three dimensions and higher is far more limited. Faced with the polynomial lower bounds of [cf:BM], the best known upper bound for dimension is for large enough (as usual being the sidelength) due to [cf:Sugimine]. Compare this with the case of no (i.e. free) boundary conditions case where it was shown in [cf:Thomas] that (and thus also ) is at least for some and an absolute constant .
In two dimensions, ever since the work of Martinelli [cf:Martinelli] in 1994 (an upper bound of at low enough temperatures) and until quite recently no real progress has been made on the original problem. Nevertheless, various variants of this problem became fairly well understood. For instance, nearly homogenous boundary conditions were studied in [cf:Alexander, cf:AY]. Analogues of the problem on nonamenable geometries (in terms of a suitable parameter measuring the growth of balls to replace the sidelength) were established, pioneered by the work of [cf:MSW] on trees and followed by results of [cf:Bianchi] on a class of hyperbolic graphs of large degrees. The SolidOnSolid model (SOS), proposed as an idealization of the behavior of Ising contours at low temperatures, was studied in [cf:MS] where the authors obtained several insights into the evolution of the contours. Finally, the conjectured Lifshitz behavior of was confirmed at zero temperature [cf:CSS, cf:FSS, cf:CMST], with the recent work [cf:CMST] providing sharp bounds also for nearzero temperatures (namely when for a suitably large ) in both dimensions two and three.
As mentioned above, the barrier was finally broken in the recent paper [cf:MT], replacing it by for an arbitrarily small and sufficiently large (where the constant diverges to as ). At the heart of the proof of the main result of that paper ([cf:MT]*Theorem 1.6) was an inductive procedure which will serve as our main benchmark here. We will shortly review that argument in §3 in order to motivate and better understand the new steps gained in the present work.
Finally, there is an extensive literature on the phaseseparation lines in the 2d Ising model, going back to [cf:AR, cf:Gallavotti]. In §2 we will review the tools we will need from the randomline representation framework of [cf:PV1, cf:PV2]. For further information see e.g. [cf:Pfister] and the references therein.
2. Preliminaries
2.1. General Glauber dynamics
The class of Glauber dynamics for the Ising model on a finite box consists of the continuoustime Markov chains on the state space that are given by the generator
(2.1) 
where is the configuration with the spin at flipped and the transition rates should satisfy the following conditions:

Finite range interactions: For some fixed and any , if agree on the ball of diameter about then .

Detailed balance: For all and ,
where .

Positivity and boundedness: The rates are uniformly bounded from below and above by some fixed .

Translation invariance: If , where and addition is according to the lattice metric, then for all .
The Glauber dynamics generator with such rates defines a unique Markov process, reversible with respect to the Gibbs measure . The two most notable examples for the choice of transition rates are

Metropolis: .

Heatbath: .
See e.g. [cf:Martinelli97] for standard comparisons between these chains, in particular implying that their individual mixing times are within a factor of at most from one another (hence our results apply to every one of these chains).
2.2. Surface tension
Denote by the surface tension that corresponds to the angle , defined as follows. Associate with each angle the unit vector and the following b.c. for :
Let be the partitionfunction of the corresponding Ising model and, as usual, let denote the partitionfunction under the allplus b.c. The surface tension in the direction orthogonal to is the limit
which gives rise to an even analytic function with period on (a closed formula appears e.g. in [cf:PV1]*Section 5). One can then extend the definition of to by homogeneity, setting , where denotes the Euclidean norm of and is the angle it forms with . For all this qualifies as a norm on .
The surface tension measures the effect of the interface induced by the boundary conditions on the freeenergy and thus plays an important role in the geometry of the low temperature Ising model. For instance, it was shown in [cf:Shlosman] that the large deviations of the magnetization in a square are governed by (also see [cf:Ioffe1, cf:Ioffe2]).
One of the useful properties of the surface tension is the sharp triangle inequality (see for instance [cf:PV1]*Proposition 2.1): For any there exists a strictly positive constant such that for any we have
(2.2) 
A thorough account of additional properties of the surface tension may be found e.g. in [cf:DKS] and [cf:Pfister].
2.3. Duality
Let denote the dual lattice to . The collection of edges of and of will be denoted by and respectively. It is useful to identify an edge with the closed unit segment in whose endpoints are , and similarly do so for edges in . To each edge there corresponds a unique dual edge defined by the condition .
Given a finite box of the form , the dual box is . The set of dual edges of , denoted by , is the set of dual edges for which both endpoints lie in . Notice that for each edge such that , the corresponding dual edge necessarily belongs to . These definitions readily generalize to an arbitrary finite , in which case consists of all dual sites whose distance from equals .
For any we associate the dual inversetemperature via the duality relation . Notice that for any the dual inverse temperature lies below which is the unique fixed point of the map . We will often refer to the Gibbs measure on a subset of the dual lattice at the inversetemperature under free boundary, denoting it by . The following wellknown fact addresses the exponential decay of the twopoint correlation function for the free Ising Gibbs measure above the critical temperature.
Lemma 2.1 (e.g. [cf:MW]*p309 Eq. (4.39), together with the GKS inequalities [cf:Griffiths, cf:KS]).
Let and . There exists some such that for any ,
A matching exponent for the spinspin correlation was established by [cf:GI] for two opposite points in the (dual) infinite strip. Let for some integer and fix . In the dual we let and and consider the free Gibbs measure at inversetemperature . It was shown in [cf:GI]*formula (2.22) that in this setting there exists some such that
(2.3) 
where the term tends to as .
2.4. Contours
Let be a finite subgraph of . The boundary of a subset of dual edges , denoted by , is the set of vertices of with an odd number of adjacent edges of . If we say that is closed, otherwise it is open.
A chain of sites of length from to in has the standard definition of a sequence of sites such that and for all . A chain from to is similarly defined with the exception that the distance requirement is relaxed into for all . A path from to in is a chain of sites consisting of edges of , that is for all . We say that a path is closed if its endpoint and starting point coincide, otherwise we say that it is open.
A set of dual edges can be uniquely partitioned into a finite number of edgedisjoint simple lines in called contours. This is achieved by repeating the following procedure referred to as the SouthEast (SE) splittingrule: When four bonds meet at a vertex we separate them along the SEoriented diagonal going through the intersection. Alternatively, one may globally apply the SW splittingrule, analogously defined with the SouthWest orientation replacing the SouthEast one (see Figure 1).
Contours can be either open or closed (with the same distinction as in paths). The length of a contour , denoted by , is the number of edges in , and the length of a collection of contours will simply be the sum of all the individual lengths. Given a finite family of contours we say that it is compatible if it is the contour decomposition of its collection of dual edges . We further say that is compatible (or compatible) to emphasize that in addition all the edges of belong to , the edgeset of .
Given boundary conditions and a box , each spinconfiguration compatible with outside (i.e. for any ) can be uniquely specified by giving all the edges such that and (that is, all edges whose endpoint sites disagree). Equivalently, one can specify the corresponding dual edges of . By applying the above contour decomposition we see that each configuration compatible with is uniquely characterized by its collection of closed and open contours (see Figure 2 for an illustration). The open contours obtained in this manner are called the phaseseparation lines.
It is clear that the boundary of the open contours belongs to and must coincide with a certain set uniquely specified by the boundary conditions (i.e. independent of the values gives to the spins of ). Notice that the cardinality of , if different from zero, must be even.
A family of closed and open simple lines is called compatible if there exists a configuration compatible with in from which is obtained in the above procedure. One can easily verify that when is a box the set of compatible contours coincides with the set of compatible contours whose boundary is equal to .
2.5. Randomline representation
For a finite subgraph of and an compatible family of contours , two different partition functions and will turn out to be useful for a given :
(2.4)  
(2.5) 
Using and we define the weight (not necessarily a probability distribution) corresponding to the family of contours , denoted by , to be
(2.6) 
The key reason for the above formula is the following randomline representation for evenpoint correlation functions: Consider the Ising model on at inverse temperature and free boundary conditions. Let be the associated Gibbs measure and let have even cardinality. Then the following holds (see [cf:PV1]*Lemma 6.9):
(2.7) 
Remark.
If the cardinality of is odd then the r.h.s. of (2.7) is zero by symmetry and the l.h.s. is zero due to the definition of .
Back to the low temperature Ising model in a box with boundary condition , let be a collection of compatible open contours. Then, by construction,
(2.8) 
where with a slight abuse of notation we have identified with the graph and in the last equality we used (2.7). The above formula will be the starting point of the proof of the new equilibrium estimates, Propositions 4.4 and 4.5.
We conclude this section with some of the main properties of the weights . For further information see [cf:PV1, cf:PV2].
Lemma 2.2 ([cf:PV1]*Lemma 6.3).
Let be a finite subgraph of and let be a family of compatible contours (open and closed). If is a subgraph of then .
Remark.
Let be a subgraph of . The edgeboundary of an edge , denoted by , is comprised of the edge itself together with any edge that is incident to it and would belong to the same contour in the contour decomposition of via the agreed splittingrule. For instance, with the SE splittingrule the horizontal edge in the dual lattice would have an edgeboundary of . Given a subset of edges we define its edgeboundary as . This definition implies that two contours and , where is closed and is either open or closed, are compatible if and only if the edgeset of does not intersect (see the related [cf:PV1]*Lemma 6.1). The following lemma is a special case of [cf:PV1]*Lemma 6.4):
Lemma 2.3 ([cf:PV1]*Eq. (6.17)).
Let be a subgraph of and let and denote two compatible families of contours with corresponding edgesets and respectively. If is compatible (or equivalently if ) then
where is the subgraph of given by the edgeset .
We will frequently need estimates on the weight of a contour constrained to go through certain dual sites; to this end, the following definition will be useful. Let and let be two open contours such that and . We say that are disjoint if either they are compatible or their edgesets are disjoint and the contour decomposition of the union of their edges is a single contour . Observe that in the latter case necessarily . For a pair of disjoint open contours we write to denote either the collection in the former case or the single contour in the latter.
Lemma 2.4 ([cf:PV1]*Lemma 6.5).
Let be a graph in the dual lattice . For any ,
In particular,
Corollary 2.5 ([cf:PV2]*Eq. (5.29)).
Let be a graph in the dual lattice . For any and any ,
Together with Lemma 2.1 the above lemma immediately implies an upper bound on the weights in mention in terms of the surface tensions and . The next lemma provides an analogous bound for the weights of closed contours going through a set of prescribed sites.
Lemma 2.6 ([cf:PV2]*Lemma 5.5 part (ii)).
Let be a graph in . Let and identify . Then
3. Inductive framework for rectangles with “plus” boundaries
In this section we outline the recursive scheme developed in [cf:MT] which, as mentioned in §1, established a significantly improved upper bound of for the mixing time on a box of sidelength with “plus” b.c. at sufficiently low temperatures.
Given (to be thought of as very small) and let
Similarly one defines the rectangle , the only difference being that the vertical sides contain now sites.
Definition 3.1.
A distribution of b.c. for a rectangle (which will be , or some translation of them) is said to belong to if its marginal on the union of North, East and West borders of is stochastically dominated by (the marginal of) the minus phase of the infinite system, while the marginal on the South border of dominates the (marginal of the) infinite plus phase .
The most natural example is to take concentrated on the boundary condition on the North, East and West borders, and on the South border.
Definition 3.2.
For any given consider the Ising model in , with boundary condition chosen from some distribution . We say that holds if
(3.1) 
for every . The statement is defined similarly, the only difference being that the rectangle is replaced by (and is required to belong to ).
With these definitions the iterative scheme developed in [cf:MT] can be summarized as follows.
Proposition 3.3 (The starting point).
For every (thus not necessarily large) there exists such that for every the statements and hold.
Remark.
Notice that the factor in front of the time is nothing but the negative exponential of the shortest side of the rectangle.
Theorem 3.4 (The inductive step).
For every large enough there exist constants such that:
(3.2) 
where
(3.3)  
(3.4) 
Remark.
In the original statement in [cf:MT] the obvious requirement of large was missing due to a typo.
Corollary 3.5 (Solving the recursion).
In turn, at the basis of the proof of Theorem 3.4, besides the so called PeresWinkler censoring inequality (see [cf:noteperes] and [cf:MT]*Section 2.4), there were two key equilibrium estimates on the behavior of (very) low temperature Ising interfaces which we now recall and which were the responsible for both the various error terms in and the constraint on the inversetemperature. The latter was necessary since the techniques of [cf:MT] were based on several results of [cf:DKS] on the Wulff construction which in turn use in an essential way low temperature cluster expansion.
3.1. Equilibrium bounds on low temperature Ising interfaces used in [cf:MT]
The first estimate is the key for the proof of the first part of the inductive statement namely . Given the rectangle write it as the union of two overlapping rectangles, each of which is a suitable vertical translate of the rectangle (see Figure 3).
Call the lowest rectangle and the highest one. Then
Lemma 3.6 (see Claim 3.6 in [cf:MT]).
There exists such that
(3.6) 
where