Hydrodynamic limit equation for a lozenge tiling Glauber dynamics
We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. . Single updates consist in concatenations of elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to , the dynamics is known to enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average  and the relaxation time of the Markov chain is diffusive , growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the distance between solutions. The mobility coefficient in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium.
2010 Mathematics Subject Classification: 60K35, 82C20, 52C20
Keywords: Lozenge tilings, Glauber dynamics, Hydrodynamic limit, Local equilibrium
The large-scale time evolution of interfaces separating different thermodynamic phases is a classical subject in statistical mechanics. A first natural goal is that of obtaining a hydrodynamic limit : take an initial interface configuration that approximates a macroscopic smooth profile, let it evolve via a microscopic Markovian Glauber-type dynamics that, at the lattice level, follows simple local rules and, rescaling time and space properly, prove that the interface converges to the solution of a deterministic PDE. If the two thermodynamic phases separated by the interface are at coexistence, i.e. if they have the same bulk free energy, we expect the correct time rescaling to be diffusive and the limit equation to be a parabolic PDE, in general a non-linear one, of the form
Here, is the surface tension functional, that is a purely equilibrium quantity, while is the (in general slope-dependent) interface mobility coefficient, that depends on the generator of the Markov chain.
To obtain a mathematically simpler model, the interface is often described at the microscopic level by a -dimensional height function (“effective interface”), i.e. the graph of a function from to (or to in the case of discrete interface models). Here, is the dimension of space where the thermodynamic system of interest lives and of course the physically most relevant case is . In the “effective interface” approximation, the internal structure of the two bulk phases is forgotten and the occurrence of interface overhangs is entirely neglected. Despite this somewhat drastic simplification, and despite the fact that the phenomenological picture behind the expected hydrodynamic limit is rather clear , most effective interface dynamics remain mathematically intractable and rigorous progress is very limited, especially for . One notable exception is that of the Langevin dynamics for the Ginzburg-Landau model with symmetric and strictly convex potential, where a rigorous derivation of the hydrodynamic limit was obtained for any by Funaki and Spohn  (see also  who extended  beyond the case of periodic boundary conditions).
Leaving aside the problem of rigorously proving the hydrodynamic limit, even the more modest goal of guessing the exact form of the limit PDE is in general out of reach, except for lucky exceptions (the Ginzburg-Landau model being one of them) where the dynamics satisfies some form of “gradient condition”  which allows to obtain a simple formula for the interface mobility , involving only equal-time equilibrium averages.
The goal of the present work is to present a Markov chain for a discrete interface model in dimension and to show that it should admit a hydrodynamic limit that is fully explicit and non-trivial ( is a non-linear function of the interface slope). We comment below on what is missing in order to turn our arguments into a rigorous proof.
Before introducing the interface dynamics we are interested in, we make a brief detour to motivate the reader. A class of discrete interface dynamics that attracted much attention lately are Glauber dynamics of dimer models, in particular lozenge tilings of the plane . Such tilings are in bijection with -dimensional discrete surfaces obtained as a monotone stacking of elementary cubes in , see Figure Figure 1.
Here, “monotone” means that the heights of columns of cubes, indexed by the coordinates of their orthogonal projection on the horizontal plane, are weakly decreasing both w.r.t. and .
The most natural reversible Markov dynamics on such tilings is the one whose elementary moves are rotations by an angle (with transition rate ) of three lozenges sharing a common vertex, see Figure 2.
This will be called the “single-flip dynamics” in the following. As discussed for instance in , the single-flip dynamics coincides with the zero-temperature Glauber dynamics of spin interfaces of the three-dimensional Ising model with zero magnetic field, where spins flip one by one. In terms of monotone stacking of cubes, the dynamics corresponds to adding/removing a cube to/from a column, with transition rate , provided the cube stacking remains monotone after the update. Recently it was proven that, if we restrict the single-flip dynamics to domains of diameter , under certain restrictions on the domain shape the mixing time is of order as . These results support the idea that the correct time-scale to observe a hydrodynamic limit should be diffusive (i.e. that we should rescale time by to see a macroscopic evolution) but they are far from being sufficient for proving convergence to a limit PDE.
In the present work, we study two modifications of the single-flip dynamics, where one allows a number of cubes to be added/removed from a column in each move, again subject to the constraint that the update is legal (i.e. that the resulting configuration is still a monotone stacking of cubes). If the rates are carefully chosen as functions of , the dynamics enjoys much nicer properties than the single-flip one. The first dynamics we will consider is the one where the transition rate of a legal update involving the addition/removal of cubes is proportional to ; in the second dynamics, instead, with rate the height of each column of cubes is resampled from the uniform distribution on all the allowed values it can take given the height of neighboring columns. See Definitions ? and ? below for more details. The former dynamics was originally introduced in , and the latter in . Both are known to satisfy the special property that the volume difference between two configuration is (on average) decreasing with time, which allows to deduce that the mixing time is at most polynomial in . Moreover, it was proven in  that the inverse spectral gap of the dynamics is and that, in special domains, a certain one-dimensional projection of the height function satisfies on average the one-dimensional discrete heat equation.
Here we show that, under a reasonable but unproven assumption of local equilibrium, one can obtain the explicit form of the hydrodynamic limit equation for the height function, see Eqs. , and below. Actually, the hydrodynamic equation turns out to be the same for both variants of the dynamics. Obtaining such an explicit expression for the hydrodynamic equation is a somewhat surprising fact; indeed, let us stress that in general (for instance, for the single-flip dynamics) the assumption of local equilibrium is not sufficient to guess the limit equation: knowledge of corrections to local equilibrium is also necessary. There is a general heuristic formula  for the mobility coefficient which is a variant of Green-Kubo formula. It is given as the sum of two terms, one involving only local averages in the stationary state of slope and the second involving a time-integral of time-space correlations in the stationary state. The latter term cannot in general be computed as it would require a closed form for space-time correlations. However in lucky cases (like ours, see Section 4) this term happens to be zero due to a summation by parts at the discrete level.
As we already mentioned, our derivation of the hydrodynamic limit relies on an unproven assumption of local equilibrium. There are various difficulties in proving such assumption, and the direct application of standard entropy techniques (see e.g. ) seems out of question, in particular because the stationary measures of the model exhibit long-range correlations. The adaptation of the so-called method employed in  looks also challenging: technically a non-trivial difficulty is to get some a-priori control of interface gradients during the evolution (see Remark ? below for more details). In  an important role in this respect was played by strict convexity of the potential, that fails in our case. However, in the case where the system has periodic boundary conditions, in a forthcoming work  we manage to overcome these difficulties and to prove rigorously the validity of the hydrodynamic limit.
The hydrodynamic equation has nice analytic features. While it is not in the form of the gradient flow with respect of a surface free energy functional, it can be written in a divergence form (cf. ) that allows to show (see Section ?) that the distance between solutions contracts with time. This is an important point in the program of rigorously proving the convergence towards the hydrodynamic limit equation, and we use this property crucially in our forthcoming work  in the periodic boundary condition setting. In fact, the idea of the method is to prove that the distance between the deterministic PDE and the randomly evolving interface stays close to zero at all times. Let us recall briefly how this works in the Ginzburg-Landau model . By an entropy production argument  one shows that at positive times the law of interface gradients is locally close to a certain equilibrium Gibbs measure with an unknown slope. The crucial point is that if the slope is “wrong”, i.e. different from that of the solution of the PDE, in which case the random interface has deviated from the deterministic evolution, the derivative of the norm turns out to be negative, which means that the evolution is immediately driven back to the deterministic one (cf. ). In turn, the mathematical mechanism behind this fact is the same as the one that guarantees that the between two solutions of the PDE contracts with time.
We will also show that the distance between solutions of the limit PDE is non-increasing, and decreases only by a boundary term (Section ?). This is the analogue of the above-mentioned average volume-contraction property of the microscopic dynamics.
Finally let us point out that the exact formula for the hydrodynamic equation leads to some striking identities involving the surface tension (see notably Eq. and the discussion in Remark ?) for which it would be very interesting to find a probabilistic interpretation.
The work is organized as follows. In Section 2 we introduce precisely the model and the dynamics. The hydrodynamic equation is given (in two different but equivalent forms) in Section 3, where we also discuss some of its properties, notably volume contraction. In Section 4 we give a first justification for the limit equation, based on linear response theory. In Section 5 instead we derive the hydrodynamic equation under a local equilibrium assumption. Finally, in Section 6 we explain how to perform some useful equilibrium computations.
2The model and the dynamics
2.1Monotone surfaces and height function
We start by defining discrete monotone surfaces.
The orthogonal projection (denoted ) of a square face of is a lozenge with angles and , side-length and three possible orientations: north-west, north-east and horizontal, according to whether the normal vector to the square face in question is , or . The projection of gives therefore a lozenge tiling of . Vertices of the lozenges are the vertices of a triangular lattice of side . We will refer to north-west oriented, north-east oriented and horizontal lozenges as lozenges of types respectively. See Figure 3.
Let be the usual orthonormal vectors of . On the plane we introduce unit vectors and correspondingly coordinates as follows: a given reference vertex (for example, the one on which the origin of projects) has coordinates and the vector (resp. ), of coordinates (resp. ) is the vector from to its nearest neighbor in direction (resp. ). That is, . Note that with this choice of coordinates, triangular faces of have side-length and not .
In order to turn the correspondence between stepped surfaces and lozenge tilings into a bijection, we impose that .
Given a discrete monotone surface , the height function is defined as follows: equals minus the height with respect to the horizontal plane (i.e. minus the coordinate) of the point that projects on , i.e. such that . Of course since we imposed . The reason for the minus sign is that otherwise the interface gradients would be given by minus the lozenge densities (see Remark ? just below), which would lead to less readable formulas later. The height function can be naturally extended to the whole plane by linear interpolation in each face of .
We will be interested in dynamics in finite domains. For let be a simply connected, bounded union of triangular faces of that can be tiled by lozenges, and let be its boundary, seen as a collection of edges of . Assume that the site where height is fixed to zero is on .
Call the set of lozenge tilings of and the generic element of . Remark that the height function on is independent of the configuration . If wished, one can imagine that is extended to a lozenge tiling of the whole plane, just by completing with a tiling of , fixed once and for all. From this point of view, can be identified with the set of monotone surfaces such that and such that the projection of restricted to coincides with .
We will assume from now on that has a scaling limit in the following sense:
To understand the condition , recall that
Observe that the argument of in is positive, since if . The function is real analytic and strictly negative in , vanishes when and its gradient diverges when is approached. The condition guarantees in particular that the cardinality of is exponentially large in , i.e. the entropy per unit area is positive.
In view of the definition of height function, one should think of as the density of lozenges of types . We however emphasize that making point-wise sense of this intuition is a delicate problem.
2.2Translation-invariant Gibbs states
It is well known  that for every there exists an unique translation invariant, ergodic Gibbs state on the set of lozenge tilings of the plane, such that the density of lozenges of type is . Such measures have the following explicit form. Let the hexagonal lattice be the dual of and color its vertices black/white in an alternate way, see Figure 4. A white vertex is given the same coordinate as the black vertex just to its right, and the coordinates correspond to the axes introduced above. There exists a natural bijection between lozenge tilings of the plane and perfect matchings of , see Figure 4.
Take a triangle with angles and let be the length of the side opposite to . Given an edge of , say that it is of type or if it is north-west, north-east oriented or horizontal and let . Then, given an integer and edges such that the white (resp. black) vertex of is (resp. ), one has 
where, if has coordinates and has coordinates ,
and the integrals runs over the torus . Note that is unchanged if all are multiplied by a common factor. In particular, if is an edge of type one has
We have seen that the projection of any discrete monotone surface gives a lozenge tiling of the plane. Horizontal lozenges will be called “particles” and will be given a label . To each particle will be associated a “vertical position” , defined as
with the coordinates of the upper corner of the particle (horizontal lozenge), as well as a “horizontal position” (or “column coordinate”)
see Figure 4. Note that if (i.e. if the column containing has the same parity as the column containing the vertex ), and otherwise.
Recalling Remark ?, observe that when the vertical coordinate of a particle changes by , there are vertices in the triangular lattice where the height changes by .
It is well known (and easy to check) that a lozenge tiling of the plane is uniquely determined by the particle positions, provided that there is at least one particle per column, which we will assume henceforth. Recall that the height function on is independent of the configuration . From the definition of height function, we deduce that for each column , the number of particles on column that are in is the same for every . Actually the whole tiling is uniquely determined (once and on are given) by the positions of the particles in .
It is also well known and easy to check that particle positions satisfy the following interlacement properties: if are two particles on the same column with and if there is no particle on column of index with then there is a unique particle (resp. ) in column (resp. ) such that (resp. ).
In the study of our interface dynamics we will need the following two definitions:
Interpretation of in terms of particles
One can give an interpretation, purely in terms of interlaced particles, to the two parameters labelling the Gibbs measures . First of all, is the density of particles in any given column. The difference corresponds to an asymmetry parameter as follows. Look at a column, say the one labelled , and call the vertical positions of its particles , ordered so that . Given particles and , let be the unique particle in column whose vertical position satisfies . Then one has
where the limit holds -almost surely, due to ergodicity of the Gibbs measure. In other words, is a measure of how much is biased away from the mid-point .
To see why holds, look at Figure 6: running along column from position to , the number of lozenges of type that are adjacent to column to its right is
and the number of lozenges of type is
The factors keep into account the fact that particle positions in column are integers and those in column are half-integers. One has then
On the other hand,
that converges to since is the number of particles in column in a segment of length . Equation then follows.
We will study two continuous-time Markov dynamics on . Both are reversible with respect to the uniform measure . We will only define the dynamics in terms of movements of particles but recall that these determine the whole tiling. In the dynamics, only particles in can evolve.
This is equivalent to a dynamics introduced by Luby, Randall and Sinclair . Let us recall that this dynamics can be used as an auxiliary process to show that the “single-flip” dynamics, where particles are instead allowed to move only to with equal rates (provided ), has a mixing time and inverse spectral gap that is at most polynomial in .
In other words, with rate each particle is redistributed uniformly among its instantaneously available positions. This dynamics was introduced in , again as an auxiliary process to analyze the single-flip dynamics.
It is immediate to see that both are reversible w.r.t the uniform measure. We will call the generators of the two dynamics. The configuration at time will be denoted and dependence on the boundary condition as well as the index , that distinguishes between the two dynamics will not be indicated explicitly.
3The limit hydrodynamic equation
Call the initial condition of the dynamics (actually is a sequence , but we drop the subscript ) and the configuration at time (recall that we do not distinguish between dynamics I and II in the notation). Assume that approximates a smooth profile, i.e. there exists satisfying such that
for every . Let for
On general grounds , one expects to concentrate around a deterministic solution in the sense that there exists some deterministic function such that, for every and ,
Furthermore, should follow a non-linear PDE of the form
where , with defined in , and is a positive function. This equation is of parabolic type, since the Hessian matrix is strictly positive definite for (positive definiteness follows from convexity of the surface tension and strict positivity follows from the fact that the determinant of equals identically , as can also be checked from below).
The positive coefficient is called the “mobility” and in general will depend on the microscopic definition of the dynamics. In particular, a priori there is no reason for it to be the same for dynamics I and II, but we will see below that the mobility does in fact coincide in the two cases.
The meaning of is that, in the diffusive scaling, the interface velocity will be given by the gradient flow associated to the surface tension functional
times a certain mobility coefficient that depends on the local slope. Note indeed that
More explicitly, one finds from
An expression for can be obtained from linear response theory. Usually (cf.  and below), such expression is given by the sum of two terms: the first involves the average w.r.t. of a local observable and the second involves the integral over time , ranging from to , of the correlations (in the stationary process started from ) between an observable at time and an observable at time . In general, it is not possible to compute the second term explicitly. In lucky cases (e.g. the zero-temperature dynamics of interfaces of the 2D Ising model or the Langevin dynamics of the Ginzburg-Landau effective interface model ) the second term vanishes due to a summation by parts. This turns out to be the case also for our dynamics.
for both dynamics I and II. Here, and are just as in Definition ?, and we have arbitrarily chosen by translation invariance.
In Section 6 we show:
The advantage of the rewriting is that turns out to be nothing but the average interface velocity for the totally asymmetric process defined in Remark ?, in the stationary measure . Namely, let the “total current” denote the number of particles that cross a fixed vertex of the triangular lattice in the time interval for the asymmetric process. Then 
where denotes average w.r.t. the stationary process started from . In  it was proven (with somewhat different notations) that
Recall that and that these three numbers give the average fraction of lozenges of types respectively under the measure . In conclusion, both for dynamics I and II, the linear response theory mobility defined as in equals
The conjectural explicit form of the hydrodynamic limit equation is then given by , together with and .
3.1Hydrodynamic equation and volume contraction
It goes beyond the scopes of the present work to investigate the existence and regularity of the solutions of . This might be a non-trivial issue due to the singularity of and when their argument approaches . In the following of this section, we will implicitly assume that the domain and the initial condition are regular enough that admits a unique classical solution that is in where we recall that the domain is closed. In the forthcoming  we explain how to extract such existence, uniqueness and smoothness statements from the existing literature (e.g. ).
It is interesting to remark that can be rewritten as follows:
(This can be checked via a direct computation, using the definition of of and the expressions for ). One can also check that the curl of the vector field is non-zero, which prevents from writing as the gradient of some function and the equation for as the gradient flow w.r.t. the associated functional .
The rewriting has two interesting consequences, namely contractions in time of both the and distances between solutions. The two phenomena are somewhat different: as we see in a moment, contraction is only a boundary effect, while contraction is a bulk effect.
By Gauss’ theorem, implies that the time derivative of the total volume below the surface, , is only a boundary term:
with the exterior normal vector to . A stronger property holds. Let be two smooth initial conditions for with
Then, one can show (see end of Section ?) that
Inequality remains true for all times, by the usual comparison principle for parabolic PDEs  (another way to convince oneself that order is preserved is to recall that the microscopic dynamics is monotone : if two initial conditions satisfy everywhere, then the two evolutions can be coupled in a way that domination is preserved at all times). One concludes that
the drift of mutual volume is a boundary effect and is negative. This fact has a microscopic analog: in fact, if are two configurations in with everywhere in , then the volume drift
(with the generator of the Markov chain) is negative and is non-zero only due to a boundary effect. This was proven in  for dynamics I and in  for dynamics II, and is actually the crucial step in the proof that the mixing time is polynomial in . In this perspective, it is natural to recover such volume decrease property in the hydrodynamic equation.
It is worth emphasizing that volume contraction is not an a-priori obvious property. In particular, it is easy to check that the single-flip dynamics, at the microscopic level, does not contract volume.
Let again be two smooth solutions of , with the same boundary data on . This time we do not require that . We have
We claim now that
whenever belong to the triangle , which implies that the time derivative in is negative. To prove , it is sufficient to prove that the matrix
is positive definite for every . The trace of is
(recall that ) while
Given that for , the ratio in the r.h.s. of is positive. As for the term , one can check that its unique extremum for is at , where .
3.2Another form for the hydrodynamic equation
There exists another way of guessing the hydrodynamic limit equation, this time not based on linear response but on a “local equilibrium” assumption. For this derivation, it is actually more convenient to use a different way of parametrizing the interface and the height function. The corresponding expression for the hydrodynamic limit equation will show an interesting link between mobility and surface tension, see .
Level set function
Let be the linear subspace of orthogonal to (i.e. the plane ). On we take coordinates whose unit vectors are the orthogonal projections of the Cartesian unit vectors of (i.e. ) and such that the point of coordinates is the projection of . Given , let
with the usual scalar product on and the unique
We will call the function the “level set function” of the interface , in order to distinguish it from the “height function” . The reason for the name is the following. Given , consider the intersection of the surface with the horizontal plane . With reference to Figure 7, each can be viewed as a simple-random walk path in space-time dimension : the time axis is horizontal, and . Moreover, these lines are mutually non-intersecting: .
It is easy to check that, modulo a global additive constant independent of , one has
Next, we define the analogue (in this new parametrization of the surface) of the domain . Given as in Section 2.1 and , let be the monotone surface whose projection is . Let and denote an arbitrary extension of to a tiling of the whole plane and the corresponding monotone surface (see discussion just before Assumption ?). We let
and we note that is independent of the choice of . Actually, on the function depends only on the arbitrary choice of outside of .
The following is equivalent to Assumption ? and is actually a rephrasing of it:
All these claims follow from Assumption ? and the change of variable formulas in Section 5.1 below; in particular, Eq. shows that is equivalent to the non-extremality condition .
Hydrodynamic limit for the level set function
Let for and
Recall that we assume that the initial condition of the dynamics satisfies . As in Proposition ?, in terms of the “level set function” this implies that there exists a smooth satisfying such that
The conjectural existence of a hydrodynamic limit means existence of a function , such that for every ,
Under a (reasonable) assumption of local equilibrium, we find (see Section 5.2) that has to satisfy the PDE
where is defined as
We will prove in Section 6:
Equation then becomes
and is defined in .
The derivatives of are a combination of trigonometric functions, but it is best to express them in terms of the surface tension. Given Theorem ?, it is easy to deduce (see Section Section 5.1) that, given satisfying , one has
Then, one can check
with . Altogether, the conjectural hydrodynamic limit equation is
A couple of comments are in order. First, as it should, equations and (together with ) are actually the same PDE, as can be seen with a suitable change of variables (see Section Section 5.1 for details). A second remark is that in the coordinate system one obtains a surprisingly simple relation between interface mobility and surface tension. Namely, rewriting (in analogy with ) the PDE satisfied by as
and comparing with one sees that the mobility coefficient is
It would be interesting to understand whether there is any thermodynamic explanation for such relation.
Finally, from one sees that the equation for is also given by