Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach
We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds for the mixing time of the chain, where is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension . Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D. Wilson  in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales ( for monotone surfaces and for the SOS model). Finally, combining techniques from kinetically constrained spin systems  together with the above mixing time result, we prove an almost diffusive lower bound of order for the spectral gap of the SOS model with horizontal size and unbounded heights.
2000 Mathematics Subject Classification: 60K35, 82C20
Keywords: Mixing time, Lozenge tilings, Solid-on-Solid model, Monotone surfaces, Glauber dynamics, Mean curvature motion.
This work was supported by the European Research Council through the “Advanced Grant” PTRELSS 228032
Understanding the dynamical behavior of interfaces undergoing a stochastic microscopic evolution and the emergence of mean curvature motion on the macroscopic scale is a fundamental problem in non-equilibrium statistical mechanics . Even the simpler question of rigorously establishing the correct time-scale for the relaxation to equilibrium is in many cases a challenge. Similar questions arise also in combinatorics and computer science when the interface configurations can be put in correspondence with the dimer coverings of a planar graph: the main focus there is to evaluate the running time of Markov Chain algorithms which sample uniformly among such combinatorial structures (cf. in particular  for background and motivations in this direction). In this paper we address this question for two natural and widely studied models and obtain essentially optimal bounds on the equilibration time.
The first classical example is the continuous-time single spin-flip dynamics of discrete monotone surfaces with fixed boundary, cf. for instance . Monotone surfaces can be visualized as a stack of unit cubes centered around the vertices of , which is decreasing in both the and the direction (see Figure 1); see Section 2 for the precise definition.
This is equivalent to the zero-temperature dynamics of the three-dimensional Ising model with boundary conditions that enforce the presence of an interface; alternatively, it can be seen as a stochastic dynamics for lozenge tilings of a finite region of the plane, or of dimer coverings of a finite region of the honeycomb lattice. The invariant measure is uniform over all discrete surfaces compatible with the monotonicity constraints and with the boundary conditions. The conjectured behavior of the mixing time is of order if is the linear size of the region under consideration. The monotonicity condition produces dynamical constraints which prevent the application of standard tools to obtain non-trivial bounds on . To overcome this difficulty, a modified “non-local” version of the dynamics, whose moves involve adding or removing piles of unit cubes stacked one on top of the other, was introduced in . Using this device, a polynomial (in ) upper bound on was proven in . An important breakthrough was obtained by D. Wilson in , where the mixing time of the non-local dynamics was sharply analysed and shown to be of order (from below and above). Via classical comparison arguments, this implies  that for the single-site dynamics - here and below we use the notation for any quantity that is bounded above by up to polylog() factors. An improved comparison, relying also on the so-called Peres-Winkler censoring inequalities , shows that ; see .
The second example is the continuous-time stochastic one-dimensional SOS model, described by a set of integer-valued heights such that each is confined in an interval whose size is of order (see Section 3 for the precise definition and Figure 2 for an illustration) and the heights are fixed boundary conditions.
The dynamics involves local moves where the discrete heights can change by at each step. The invariant measure is the Gibbs distribution corresponding to a potential given by the absolute value of the height gradients. Due to the non-strictly convex character of the interaction, even obtaining a diffusive spectral gap bound (of order ) for zero boundary conditions has been a long-standing open problem. The analysis of an auxiliary non-local dynamics, in the spirit of , plus a judicious use of the Peres-Winkler inequalities were recently combined to obtain a mixing time upper bound , while again the conjectured behavior is .
The main contribution of the present paper is a proof that, for both problems, ; see Theorem ? for monotone surfaces and Theorem ? for SOS. In the monotone surface case, we must restrict our analysis to the case where the boundary condition is approximately planar (see e.g. Figure 1), so that on a macroscopic scale, at equilibrium, the surface is flat, cf. Theorem ?. At the microscopic scale this corresponds to the zero temperature limit of the 3D Ising model with so-called Dobrushin boundary conditions, i.e. the boundary spins take the value according to whether they lie above or below a given plane. In any finite volume, this leads to the uniform distribution over all monotone surfaces compatible with the given “planar” boundary conditions. In the infinite volume limit , such uniform distribution is related to a translation invariant, ergodic Gibbs measure on dimer coverings of the infinite honeycomb lattice , described by a determinantal point processes whose kernel is known explicitly. Our method relies crucially on the Gaussian Free Field-like fluctuation properties of such infinite volume state.
Theorem ?, together with techniques developed in the context of kinetically constrained spin models , allows us to implement a recursive analysis whose output is an almost diffusive lower bound up to polylog() factors for the spectral gap of the SOS model with unbounded heights and zero boundary conditions ; see Theorem ?.
Our approach can be roughly described as follows. At equilibrium the interface is macroscopically flat, with maximal height fluctuations much smaller than (logarithmic in the monotone surface case and of order for SOS). The main step in the proof of the mixing time upper bounds is to show that an initial macroscopically non-flat profile approaches the flat equilibrium profile within the correct time. Heuristically, the interface evolves by minimizing the surface tension and therefore feels a drift proportional to the local mean curvature. A major difficulty encountered in previous approaches is the possible appearance of very large (up to order ) gradients of the surface height. Our method consists in introducing an auxiliary “mesoscopic” dynamics which approximately follows the mean curvature flow, such that large gradients are absent in the initial condition and are very unlikely to be created at later times. The key point is that, thanks to monotonicity and coupling considerations, the original dynamics converges to equilibrium faster than the auxiliary one.
The study of the mesoscopic dynamics involves a local analysis of the relaxation of a mesoscopic portion of the interface, whose size depends i) on the current local mean curvature of the surface at time and ii) on the size of equilibrium height fluctuations. For the monotone surface case it turns out that one has to choose and for the SOS model (in both cases modulo polylogarithmic factors). We refer to Section Section 4 for a more detailed explanation of the strategy which leads to the choice of the different scales in the two models. Note that, since the evolution tends to a profile with vanishing mean curvature, grows with time and becomes much larger than the initial value when equilibrium is approached. One key input is to prove that the equilibration time within such mesoscopic regions is of the correct order : this result can be obtained following recent important progress in  for monotone surfaces and in  for SOS (both use the Peres-Winkler inequalities and Wilson’s argument  in an essential way). The other key point, which is the main contribution of this work, is to show that the mesoscopic dynamics is dominated with high probability by the deterministic mean curvature evolution of a macroscopic smooth profile with the appropriate boundary conditions.
In this paper we do not focus on mixing time lower bounds. Let us however mention that, for the monotone surface dynamics, under natural assumptions mentioned in Remark ? below, one can apply  to obtain a lower bound of order for the mixing time. Similarly, for the SOS model it is known that the spectral gap is at most of order , which by standard inequalities directly implies a lower bound of order for .
We believe our method could potentially work for a wide range of stochastic interface dynamics models where mean curvature motion is expected to occur macroscopically. An example that comes naturally to mind is the dynamics of domino tilings of the plane, for which at present only non-optimal polynomial upper bounds on are available .
A real challenge is on the other hand to prove that for the monotone surface dynamics when the boundary height is not approximately planar, in which case the equilibrium shape is not macroscopically flat and arctic circle-type phenomena can occur . While in principle our idea of mesoscopic auxiliary dynamics could be adapted to this case too, what is missing here are precise, finite- equilibrium estimates on height fluctuations and on the rate of convergence of the equilibrium average height to its macroscopic limit.
Concerning the SOS model, a big challenge is the analysis of the dynamics for the model in dimension , for which not even crude polynomial bounds on are available, while on general grounds one can expect
1.1Generalities and notation
Let us recall some standard definitions for continuous time reversible Markov chains (see e.g. ). We will mostly work in the case where the state space is finite and the Markov chain is irreducible. In particular, there is a unique reversible invariant measure . For and , denotes the law of the configuration at time started from the initial configuration . The law of the chain is denoted by .
Given two laws on , we let denote their total variation distance. The mixing time , defined as
measures the time it takes for the dynamics to be close in total variation to equilibrium, uniformly in the initial condition. It is well known that
i.e., the worst-case variation distance from equilibrium decays exponentially with rate .
If denotes the infinitesimal generator of the reversible Markov chain, the spectral gap is defined as the lowest nonzero eigenvalue of . Equivalently, if denotes the associated Dirichlet form, one has
where stands for the variance and the infimum ranges over all functions such that . This definition makes sense also in the case where is countably infinite, as will be the case for the unbounded SOS model to be considered in Theorem ? below.
Throughout the paper, we will adopt the following conventions:
if , then denotes their Euclidean distance;
if , then we write for its components;
if , then denotes its diameter;
if , then . If on the other hand is a smooth subset of , then denotes its usual boundary.
2Monotone surfaces with “planar” boundary conditions
On there is a natural partial order: we say that if for every . Analogously, for we write if for every .
2.1Heat bath dynamics
We define a dynamics on monotone surfaces with initial condition and fixed boundary conditions (b.c.) outside a finite region. Let be a finite connected subset of (the finite region) and (the boundary condition). Without loss of generality we will always assume that contains the origin.
Given and , let be defined by
The dynamics is a continuous-time Markov chain on the set
The initial condition at time zero is some given . To each is assigned an i.i.d. exponential clock of rate . If the clock labeled rings at time , we replace with or with equal probabilities. It is immediate to check that such Markov chain is irreducible and reversible with respect to the uniform measure on , which we denote or simply . The mixing time is then defined as in where the supremum is taken over .
A function on is said to be increasing (resp. decreasing) if (resp. ) whenever . Given two laws on , we write ( dominates stochastically ) if for every increasing function . The heath-bath dynamics is monotone (or attractive) with respect to the partial ordering “”, in the following sense. If denotes the law of , the dynamics at time started from and evolving with b.c. , one has the following property (cf. for instance the discussion in ):
In particular, letting , one has . It is possible to realize on the same probability space the trajectories of the Markov chain corresponding to distinct initial conditions and/or distinct boundary conditions in such a way that, with probability one,
Such a construction takes the name of global monotone coupling. Throughout the paper we will apply several times the above monotonicity properties: for brevity, we will simply say “by monotonicity...”
2.3Mixing time upper bound
As we mentioned in the introduction, it is expected that , where is the diameter of the region . The next result proves such conjecture, up to logarithmic corrections, under the assumption that the boundary conditions are “approximately planar” (cf. condition below). Such “planar” case is rather natural in terms of the three-dimensional Ising model: indeed, it corresponds to the zero-temperature limit of a system defined in the cylinder , with Dobrushin-type boundary conditions which are, say, “” above some plane and “” below.
The planarity condition on the boundary conditions is specified as follows:
As we mentioned in the introduction, under such boundary conditions the surface at equilibrium is essentially flat:
We can finally formulate our mixing time upper bound:
With some technical effort (but no need of new ideas) one can improve the exponent to but we will not do so, since neither is close to the conjectured optimal value .
2.4Dynamics with “floor” and “ceiling”
In the course of the proof of Theorem ? we need an auxiliary restricted dynamics for an interface constrained between a floor and a ceiling. Let and be as in the previous section; fix some with and let
One can define a dynamics restricted to simply by choosing an initial condition and redefining
(compare with Eqs. , ). The dynamics is again monotone in the sense of Section 2.2, but this time the invariant measure is the uniform measure on .
3 Solid-on-Solid model
We turn to the study of the mixing time and spectral gap of a one-dimensional interface of Solid-on-Solid (SOS) type. The generic configuration (height function) of the standard SOS model is and its equilibrium measure corresponding to boundary conditions is
with and . There is no inverse temperature parameter in since in this one-dimensional model its numerical value does not affect the qualitative behavior of the system and there is no loss of generality in fixing its value to unity. It is well known that describes the law of the unique open contour in the two-dimensional Ising model in the box with Dobrushin boundary conditions (boundary spins are “” under the line which joins to and “” below it), in the limit where the couplings on vertical edges tend to infinity.
Since the mixing time deals with relaxation to equilibrium from an arbitrary initial condition it is necessary to introduce the following bounded version  of the SOS model, enclosed in a rectangular box of sides of order . Thanks to standard equilibrium estimates, see also Lemma ? below, the behavior at equilibrium of this bounded version of the model is essentially the same as the usual unbounded one defined above. We come back to the unbounded model in Theorem ?, which deals with the spectral gap.
For nonnegative integers and , consider the configuration space defined by
The equilibrium measure on is then given by .
Occasionally we will consider the SOS model with further hard wall constraints, obtained by conditioning to the event , where , are two configurations such that . Here, and below, we use the notation , for the natural partial order in defined via , for all . We refer to as the floor and the ceiling, respectively, and write for the corresponding equilibrium measure. If denotes the maximal configuration in , i.e. , we sometimes consider the model with and where for all , i.e. the interface is above the straight line connecting the two boundary values, cf. Figure 2 (b). In this case one speaks simply of an interface above the wall. Note that is the SOS equivalent of the “monotone surface with fixed slope”, cf. Definition ?.
In the following, whenever we do not explicitly mention floor and ceiling, it is understood that we are talking about the bounded model where and , where is the minimal configuration in : .
The evolution of the interface is given by the standard heat bath dynamics, i.e. single-site Glauber dynamics described as follows. There are independent Poisson clocks with mean at each site . When site rings, the height is updated to the new value or with probabilities , respectively, determined by:
where and . With the remaining probability , stays at its current value. It is not hard to check that this defines a continuous time Markov chain with state space and stationary reversible measure given by . In the sequel we will write for the random variable describing the state of the Markov chain at time with initial state and for its distribution. Let denote the mixing time of this Markov chain.
We may consider the evolution of the system under hard wall constraints as above. This amounts to the same dynamics except that any update which would violate the constraints is rejected. The dynamics is then reversible w.r.t. the equilibrium measure associated to the floor and the ceiling , see also Section 2.4 above for the analogous constrained dynamics in the monotone surface case. In either case, with or without hard walls, the monotonicity considerations recalled in Section 2.2 apply here without modifications, with the natural partial order on configurations introduced above.
Our main result about the mixing time of the SOS interface is:
The unbounded version of the SOS model is given in . We write again for the corresponding Gibbs measure. The dynamics is the same as above except for the absence of the constraints , i.e. when the clock labeled rings, is updated to the new value with probability given by . The infinitesimal generator is given by
where , and is the configuration coinciding with everywhere except that at site the value of is replaced by . The Dirichlet form is given by
where denotes expectation w.r.t. . Note that, for each finite , defines a bounded self-adjoint operator in . The associated spectral gap is defined by , where ranges over all with nonzero variance.
This estimate is optimal, modulo the logarithmic factor: an upper bound is given e.g. in . The lower bound was proven in  for a modified version of the SOS model with weak boundary couplings; such modified model is much less sensitive to the boundary conditions and has a genuinely different dynamical behavior. If instead the absolute value interaction potential were replaced by one with strictly convex behavior at infinity, then the correct lower bound would follow by well-established recursive methods, see e.g. .
4Strategy of the proof
As already announced in the introduction, despite the fact that the equilibrium fluctuations of the interface in the two models are very different, our bound is proved following a common strategy that we sketch here.
The crucial step is the following (see Propositions ? and ? below for a precise formulation in the case of monotone surfaces and SOS model):
At that point one can conclude provided that a result of the following type is available (see e.g. Proposition ? below in the case of the SOS model):
In turn, Step ? follows if one proves that, with high probability, the interface started from the maximal configuration stays below a deterministic interface evolution which after time is at the correct distance from the flat profile. It turns out that it is actually sufficient to define the deterministic interface evolution along a sequence of deterministic times . At all times , the deterministic interface is the boundary of where, given , is a spherical cap (if we are considering two-dimensional interfaces like monotone surfaces) or a circular segment (in the case of one-dimensional interfaces like SOS) of height and base of linear size roughly of order , see Figure 3. The base of lies on the plane/line which contains the macroscopic flat profile. The evolution of , by a kind of “flattening process”, in the time interval transforms into . The sequence of increasing times and of decreasing heights will be introduced in a moment. The “domination statement” then is of the following type (see Propositions ? and ?):
The initial height is taken to be proportional to and one sets ; this guarantees that the statement of Claim ? holds trivially for . In order to choose given , one uses the following procedure. Consider the spherical cap/circular segment and choose a point on its curved boundary (e.g. the highest one). Move inward (i.e. inside ) the tangent plane/line at the chosen point by an amount and call the diameter of the intersection between the plane/line with , see Figure 3.
Then is chosen as the critical value such that the equilibrium fluctuations on scale are of order (apart from logarithmic corrections), i.e. . Also, is the smallest index such that , i.e. is of the order of the equilibrium height fluctuations on scale . As for the time sequence , one sets to be of order (again neglecting logarithmic corrections): that this is the correct choice is guaranteed by a careful use of Step ?, applied with . It is not difficult to realize that . Indeed, assume for definiteness that for some , where if we mean that . Then, simple geometric considerations show that
Approximating the recursion for with a differential equation gives
since both and are of order . In particular, one has roughly . Then,
since the last sum is of order . Remarkably, the order of magnitude of does not depend on the fluctuation exponent , while the sequence and the value of do. The statement of Claim ? for allows to conclude Step ?: the evolution started from the maximal configuration, at time , is below the deterministic evolution, which is within distance from the flat profile.
Another way to understand the choice of the time-scales is the following. If one imagines that the boundary of evolves by “mean curvature”, i.e. feeling a inward drift proportional to the inverse of its instantaneous radius of curvature, then the time to transform into must be , where is the radius of curvature of . One can easily check that, apart from logarithmic corrections, this coincides with the requirement .
5Monotone surfaces: Proof of Theorem
An essential tool in the proof of Theorem ? are the translation invariant, ergodic Gibbs measures, with given slope , on the set of monotone surfaces . The trick of using the properties of such infinite-volume states to obtain fluctuation bounds for surfaces with fixed boundary conditions around a finite region was already crucial in . The most relevant result for this purpose is the the following.
Concerning height fluctuations under these Gibbs measures one can prove that for every there exists a positive constant such that, for every and large enough,
(see ; the proof is given there for and but it works identically for ). In other words, the surface lies on average on a plane of slope and the height difference between two points does not differ from the average height difference by much more than the logarithm of . The estimate is obtained via the well-known fact  that height differences between two points can be written as the number of points of a determinantal point process whose kernel is explicitly known.
The connection between the measure and the “infinite volume states” for dimer coverings of the infinite honeycomb lattice defined in  is well known and is discussed for instance in . Let us just recall that the components are directly related to the fractions of dimers of the three possible types (horizontal or rotated by or by ).
We now turn to the proof of . By symmetry it is enough to show that
By monotonicity the probability in the left-hand side of is increased whenever is replaced by such that . In particular, let be sampled from where is a given element of , is such that and is the event . From and the fact that one sees that
This is because, if , implies , recall Definition ?, so that . Therefore, the probability in the left-hand side of is upper bounded by
where in the first step we used the DLR property and in the second one the fluctuation bound , together with the fact that with our choice of one has .
6Monotone surfaces: Proof of Theorem
In this section the slope , the region and the good planar boundary condition with slope are fixed as in Theorems ?, ?. Denote (resp. ) the maximal (resp. minimal) configuration of with respect to the partial ordering “” and recall that denotes the configuration at time started from .
The first key ingredient is a result saying that, after time of order , the surface is at most at distance away from the plane of slope (cf. Definition ?). It is here that the new ideas of mimicking the evolution by mean curvature play a crucial role.
The second result says that once the surface is within distance from , it does not go much farther than that for a time much longer than . This second step is much more standard and its proof combines monotonicity and reversibility together with the fluctuation bounds of Theorem ?.
Finally, the last step shows that if the surface evolves constrained between a ceiling and a floor which are within distance from , then mixing occurs within a time .
We can now easily put together Propositions ? to ? to obtain the desired upper bound on the mixing time:
It is a standard fact that
(see e.g.  for a similar statement) so it is sufficient to prove that
for . Let us consider e.g. the case of the maximal initial condition , the other case being analogous.
Define and, for ,
Let be a subset of . Then, using Proposition ?,
Next, from Proposition ?, one has for every
where denotes the law of the dynamics restricted to the set . Indeed, up to the random time the two dynamics and can be perfectly coupled so that they coincide. In particular, has the same law under and . Finally, thanks to Proposition ?, is at least times the mixing time of the restricted dynamics (which is ). Therefore, from and the fact that the invariant measure of the restricted dynamics is , one has
Thanks to Theorem ? one has and finally
for every event , which implies for large enough.
As a warm-up, we start by proving the easier Proposition ?.
Let satisfy . By monotonicity, if and are such that and , then
where denotes the evolution with boundary condition (instead of ). In particular, this is the case if we set and is sampled from the measure , where is the event . From Theorem ? (applied with ) one sees that
for some . This is because is within distance from the plane (cf. Definition ?) while is within distance from the plane . Therefore, the probability in is upper bounded by
The initial condition in is sampled from , which is the invariant measure of the dynamics , so that the distribution of coincides with at all later times. Via a union bound over times and recalling the relation between and , the first term in is upper bounded by
which is of order , see Theorem ?. The factor is just the average number of Markov chain moves within time , since there are order of lattice points in . Similarly one bounds the probability that for some and claim is proven.
We prove only Eq. since is obtained essentially in the same way. Let be a disk of radius
on the plane of slope (cf. Definition ?, with the same constant as in ) such that its projection on the horizontal plane contains and moreover the distance between and is at least (recall that has diameter ).
Given , let be the spherical cap whose base is the disk and whose height is . The radius of curvature is related to and by
and, since we will always work under the condition , we have . For a point on the curved portion of the boundary of , let be the normal at directed towards the exterior of . It is clear that, if , one has ; in particular, with the convention of Definition ?. Finally the height (w.r.t. the horizontal plane) of the spherical cap at horizontal coordinates is denoted by
We now define a sequence of spherical caps with constant base , decreasing height and increasing radius of curvature . More precisely, let and . Then we let and . For later purposes we also define
Recalling that , where is small uniformly in , it is immediate to deduce that
With this notation the key step is represented by the next Proposition.
Since , Proposition ? together with imply the desired inequality .
We prove the claim by induction on . For this is trivial since we chose such as to guarantee that the maximal configuration is below the function .
Assume the claim for some . For define the event
so we need to prove . We have