A (2+1)-dimensional growth process with explicit stationary measures

A -dimensional growth process with explicit stationary measures


We introduce a class of -dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. “Irreversible” means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes.

When the dynamical asymmetry parameter equals zero, the infinite-volume Gibbs measures (with given slope ) are stationary and reversible. When , are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point grows linearly with time with a non-zero speed: while the typical fluctuations of are smaller than any power of as .

In the totally asymmetric case of and on the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” introduced by A. Borodin and P. L. Ferrari in [4]. For a suitably chosen, “integrable”, initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale , exploiting the fact that in that case certain space-time height correlations can be computed exactly. In the same setting they proved that, asymptotically for , the local statistics of height fluctuations tends to that of a Gibbs state (which led to the prediction that Gibbs states should be stationary).

2010 Mathematics Subject Classification: 82C20, 60J10, 60K35, 82C24
Keywords: Interface growth, Interacting particle system, Lozenge and domino tilings, Hammersley process, Anisotropic KPZ equation

This work was partially supported by the Marie Curie IEF Action “DMCP- Dimers, Markov chains and Critical Phenomena”, grant agreement n. 621894


To motivate the object of our study, let us start with a well-known -dimensional growth process. At all times , the configuration is an integer-valued height function with space increments , see Figure 1. Local minima turn to local maxima with rate (this corresponds to deposition of elementary squares) and local maxima to local minima with rate (evaporation of elementary squares). If positive interface gradients are identified with “particles” and negative gradients with “holes”, this process is equivalent to the one-dimensional Asymmetric Simple Exclusion process (ASEP).

Figure 1: The ASEP: squares are deposed (i.e. minima turn to maxima) with rate p and evaporate (i.e. maxima turn to minima) with rate q. In the particle interpretation, particles jump to the right/left with rate q/p and cannot occupy the same site.
Figure 1: The ASEP: squares are deposed (i.e. minima turn to maxima) with rate and evaporate (i.e. maxima turn to minima) with rate . In the particle interpretation, particles jump to the right/left with rate and cannot occupy the same site.

The study of this and similar stochastic growth processes in dimension witnessed a spectacular progress recently, especially in relation with the so-called KPZ equation, cf. e.g. [27] for recent reviews. Some of the basic questions that were solved for certain models include the identification of the translation-invariant stationary states (for ASEP, these are simply the combinations of Bernoulli measures for any intensity ), the determination of the dynamic scaling exponents characterising the space-time correlation structure of height fluctuations, the study of the limit rescaled fluctuation process and its dependence on the type of initial condition. The same KPZ scaling relations appear also in the context of -dimensional directed polymers in random environment, last passage percolation and random matrix theory, just to mention a few instances [27].

On the other hand, for -dimensional stochastic growth models, , the situation is much more rudimentary and mathematical results (see notably [26]) are rare. In this work we introduce a -dimensional stochastic growth process, for which we study the stationary measures and the corresponding large-time behavior of height fluctuations. The two-dimensional interfaces entering the definition of our process are discrete (i.e. heights are integer-valued) and are given by the height function associated to dimer coverings (perfect matchings) of either the infinite hexagonal or infinite square lattice [21]. Height functions corresponding to dimer coverings of bipartite planar graphs, or to the associated tilings of the plane, are classical examples of discrete two-dimensional interfaces. For instance, dimer coverings of the hexagonal lattice (i.e. tilings of the plane by lozenges of three different orientations) correspond to discrete monotone surfaces obtained by stacking unit cubes, see Figure Figure 2. “Monotone” means that if we let denote the height w.r.t. the horizontal plane of the vertical column of cubes with horizontal coordinates , then . In a sense, discrete monotone height functions are the most natural -dimensional analogue of the -dimensional height functions appearing in the one-dimensional ASEP.

Given a density vector with , there exists [22] a unique infinite-volume translation-invariant ergodic Gibbs measure such that

  • the three types of lozenges have densities and

  • conditioned on the tiling configuration outside a finite region of the plane, describes a uniformly random tiling of .

The measures have an explicit determinantal structure that will play a role in this work and that is recalled in Section 2.2.

Figure 2: A lozenge tiling of a portion of the plane, or equivalently a monotone stacking of unit cubes
Figure 2: A lozenge tiling of a portion of the plane, or equivalently a monotone stacking of unit cubes

To model a growth process, we want to introduce a Markov evolution which is asymmetric or irreversible, in the sense that the interface has a net drift, proportional to an asymmetry parameter . Moreover, as discussed in Section 1.1 below, in order that its fluctuations can be at least heuristically described by a -dimensional KPZ-type equation, the average interface speed should be a non-linear function of the interface slope. The most natural -dimensional generalization of the ASEP described above (but which is not the one we will study here) would be the following. Let

and observe that (resp. ) is the maximal number of cubes we can add to (resp. remove from) column while respecting the condition for every . For every column , we add a single cube with rate if and remove a single cube with rate if . In words, single elementary cubes are deposed (Figure 4 top) with rate and removed (Figure 4 bottom) with rate (compare with Figure 1). We refer to this as the “single-flip dynamics”. If there is no drift and the infinite-volume Gibbs measures [22] are stationary and reversible. If instead , the stationary states are not known, but they appear to be definitely very different from the equilibrium Gibbs measures [15]. This process has been studied numerically and one finds that typical interface fluctuations grow with time like , with [15]. This is in sharp contrast with the ASEP, where the Bernoulli measures are stationary, irrespective of being equal or different from . In the language of Section 1.1, the two-dimensional single-flip growth process is believed to belong to the so-called isotropic -dimensional KPZ class when . Unfortunately, the single-flip process is very hard to analyze mathematically and very little is known rigorously.

In this work we study, instead of the single-flip dynamics, a different -dimensional irreversible growth process, that we call “bead dynamics” for reasons that will be clear later (in the hexagonal lattice case, “beads” or “particles” correspond to horizontal lozenges as in Figure 2). As discussed in Section 1.1, the bead dynamics belongs (in contrast with the single-flip dynamics) to the so-called anisotropic -dimensional KPZ class when . Updates of the dynamics consist in adding or removing a random number of cubes at some column , in the following way (see Section 2.3 for a precise definition and Section 3.1 for the analogous construction on the square lattice). For every column , we assign

  • rate to the update for every (deposition of cubes to column );

  • rate to the update for every (removal of cubes from column ).

If again there is no drift and the measures [22] are stationary and reversible. Somewhat surprisingly, turns out to be stationary (but not reversible!) for any density vector and for any value of . This is the content of our first result, Theorem ?. The same then clearly holds also if we add to the generator of the bead dynamics the generator of another process w.r.t. which is reversible. The measures and their convex combinations are the only stationary measures that can be obtained as limits of stationary measures for the bead dynamics periodized on the torus of side . In principle our result does not exclude the existence of other stationary measures that cannot be obtained this way; there might exist for instance analogs of the so-called “blocking measures” of one-dimensional asymmetric exclusion processes [13].

We emphasize that it is a non-trivial fact that equilibrium Gibbs measures should remain stationary in presence of dynamical irreversibility. As we mentioned above, this is false for instance for the single-flip dynamics. Typically, one expects that a Gibbs measure of a reversible dynamics remains stationary after introduction of a drift only when the reversible dynamics satisfies a so-called “gradient condition” [30]. As we discuss in Section ?, for the symmetric dynamics with one can indeed identify a certain “gradient condition” that might help explain why Theorem ? holds.

It is important to emphasize that stationarity of the Gibbs measures means that, if the process is started from the distribution , the law of interface gradients is time-invariant. However, overall the height function has a time-dependent random shift where, say, is the origin of the plane. On average grows like for some non-zero and slope-dependent but the amplitude of its fluctuations cannot be deduced immediately from the stationary gradient measure . Our second result, Theorem ?, says that the typical fluctuations of grow slower than any power of . Under a certain (technical) restriction on the interface slope, we can actually prove that fluctuations are at most of order , which we believe to be the optimal order of magnitude. Recall that, in sharp contrast, for the single-flip dynamics fluctuations were observed numerically [15] to grow like a non-trivial power of .

A word about Theorem ? (stationarity of ). Checking stationarity is easy for the process periodized on the torus of size , see Section 4. The extension to the infinite lattice is, however, non-trivial. One may expect that, when is large, on local scales and for finite times the system does not feel the periodic boundary conditions and therefore locally the dynamics on the torus and on the infinite lattice could be coupled with high probability. The situation is however more subtle: while on the torus the process is always well-defined, in the infinite systems one can easily construct initial configurations such that, for instance, beads (horizontal lozenges) escape instantaneously to infinity. This is due to the fact that we allow for an unbounded amount of cubes to be deposed/removed at a time, since is not bounded. In order for the coupling to work, one needs to prove that for typical initial conditions and with high probability, the random variables remain sufficiently tight in time during the out-of-equilibrium evolution. An important ingredient in overcoming these difficulties is the work [28] by Seppäläinen on the one-dimensional Hammersley process [1]. In fact, viewing beads as particles, the bead dynamics can be seen as a two-dimensional generalization of the Hammersley process, or more precisely an infinite collection of interacting Hammersley processes, see Figure 5 (a different two-dimensional generalization of the Hammersley process was introduced by Seppäläinen in [29]: in that case a full hydrodynamic limit was obtained, but the stationary measures and the size of height fluctuations remain unknown). As a side remark, the single-flip dynamics can be instead visualized in a natural way as an infinite collection of mutually interacting one-dimensional ASEPs, see caption of Figure 5.

As we explain in some more detail in Section 3, in the totally asymmetric case and on the hexagonal lattice, the bead dynamics is the same as the interacting driven particle system introduced by A. Borodin and P. L. Ferrari in [4]. In [4], for a specific, deterministic initial condition, the hydrodynamic limit and the convergence of height fluctuations on scale to a Gaussian field were obtained. For such initial condition, the above-mentioned problem of proving that the dynamics is well-posed does not arise, simply because each bead has a deterministic, time-independent maximal position it can possibly reach, and therefore cannot escape to infinity. As we mention in Section 3, on the basis of [4] it was natural to conjecture our Theorem ?.

1.1Isotropic and anisotropic KPZ classes

In order to predict whether the fluctuations of a -dimensional growth process should be described by a KPZ-type equation, one should look at the Hessian of , the average interface velocity considered as a function of the interface slope. Indeed, the evolution of the fluctuations in the stationary state of slope should be governed on large space-time scales by a stochastic PDE of the type

with a diffusion coefficient and a quadratic form whose corresponding symmetric matrix is proportional to the Hessian of at . (At present, it is not known how to regularize such equation in order to make it mathematically well-defined, as was done recently for its one-dimensional analog [18]).

The growth model is said to belong to the “anisotropic -dimensional KPZ class” when the two eigenvalues of the quadratic form have opposite sign, and to the “isotropic -dimensional KPZ class” when they have the same sign. As discussed in [5], the bead dynamics belongs to the anisotropic class (the eigenvalues can be computed explicitly from formula below for ).

Models in the anisotropic class are in a sense easier than those in the isotropic class. Indeed, in the former case it was predicted by Wolf [33] that the non-linearity is irrelevant as far as the large-time behavior of the interface roughness is concerned, i.e. the fluctuations of should be of the same order as for the linear Edwards-Wilkinson equation [12], where is set to zero. Theorem ? and Eq. confirm this prediction, for the bead model. Apart from the bead dynamics we study here, there are a few other -dimensional stochastic growth model models known to be in the anisotropic KPZ class, and all of them are exactly solvable in some sense. In this respect, let us mention the model introduced by Prähofer and Spohn in [26], for which height fluctuations are also known to grow like . See also [3] for growth models in the same universality class: it would be interesting to see whether our result extend to these processes.

The situation is very different for models in the isotropic KPZ class. In this case there are, to our knowledge, no exactly solvable models and only numerical simulations are available (see [19] for an overview). The non-linearity is expected to be relevant and to produce a non-trivial dynamical height fluctuation exponent. In particular, while neither the interface velocity nor the stationary states of the -dimensional single-flip dynamics can be computed explicitly, the model is widely believed to belong to the isotropic KPZ class and, as mentioned above, the dynamical fluctuation exponent is numerically estimated to [15].

2Irreversible lozenge dynamics and stationarity of Gibbs states

2.1Configuration space

The Markov process we are interested in lives on , the set of dimer coverings (perfect matchings) of the hexagonal lattice , or equivalently the set of lozenge tilings of the whole plane. See Figure 3.

Figure 3: A portion of dimer covering of \mathcal H and the corresponding lozenge picture. The two beads in column \ell are encircled and the vertices {\rm b}_{0},{\rm w}_{0} are marked.
Figure 3: A portion of dimer covering of and the corresponding lozenge picture. The two beads in column are encircled and the vertices are marked.

The “elementary moves” of the dynamics consist in rotating by an angle three dimers around a hexagonal face, see Figure 4.

Figure 4: The two elementary moves.
Figure 4: The two elementary moves.

In this move, a horizontal dimer moves up or down a distance . The generic move of the dynamics (defined precisely in Section 2.3), that was described in the introduction as the deposition/removal of cubes, can be seen as a concatenation of a random number of elementary moves in adjacent hexagons in the same vertical column. We can therefore see each “horizontal dimer/lozenge” (we call them “beads” hereafter1) as attached to a “column” (an infinite vertical stack of hexagons): the bead can move up and down along the column but not change column. The set of possible bead positions can be identified with on, say, even columns and with on odd columns. Note that beads of neighboring columns are interlaced: if on column there are two beads at positions then necessarily in column there is a bead at a position and similarly for column .

2.2Height function

On , we take a coordinate frame where the axis forms a clockwise angle with the usual horizontal axis and the axis an angle , see Figure 3. We also set to be the vertical unit vector.

Note that, with this convention, corresponds to minus the height function with respect to the horizontal plane, and observe also that when moving one step in the direction, decreases by if no dimer is crossed and stays constant otherwise.

Given with , and (we call a non-extremal slope) there exists a unique translation-invariant ergodic Gibbs state with slope . This is a translation invariant probability law on the set of dimer coverings of , that satisfies (cf. [21]):

  • is ergodic with respect to translations by ;

  • it satisfies the Dobrushin-Lanford-Ruelle equations: conditionally on the dimer configuration outside a given finite subset , is the uniform measure over all dimer coverings of compatible with , i.e. such that is a dimer covering of ;

  • it has slope , i.e. .

Note that is the density of south-east oriented lozenges, is the density of north-east lozenges and the density of horizontal lozenges. The non-extremality requirement on means that all three types of lozenges have non-zero density.

The measure is in a sense completely known and has a determinantal representation, that we recall here briefly (cf. in particular ), since it will be needed in the following. See [22] for further details. First of all, color sites of white/black according to whether they are the left/right endpoint of a horizontal edge and let be the sub-lattice of white/black vertices. We denote the black/white vertices indicated in Figure 3 and we let , with , be the translation of by .

Take a triangle with angles and let be the length of the side opposite to . Define the Kasteleyn matrix as follows: If are not nearest neighbors, then . If they are nearest neighbors, then or or according to whether the edge is oriented south-east, north-east or horizontal.

Define also the matrix as

where the integral is taken over the two-dimensional unit torus . The long-distance behavior of is precisely known [22]: since the polynomial has two simple zeros on the torus, decays like the inverse of the distance so that in particular

with (this in general fails if is extremal, e.g. if only one of the three dimer orientations has positive density).

Given a set of (not necessarily horizontal) edges of , the correlation function (with the indicator function that there is a dimer at ) is given by

Note, also in view of formula , that the r.h.s. of is invariant if we multiply all by a common factor , so that we may for instance fix the sum to .

2.3Definition of the dynamics and stationarity of Gibbs states

The dynamics is informally defined as follows (cf. Figure 5). To each column and to each possible bead position (horizontal edge of ) we associate two independent Poisson clocks of mean and respectively. We call them -clocks and -clocks, with obvious meaning. Clocks at different locations are independent. When a -clock (resp. a -clock) at rings, if is occupied by a bead we do nothing. Otherwise, we look at the highest (resp. lowest) bead (if any) on column that is at position lower (resp. higher) than : if it can be moved to without violating the interlacing constraints then we do so, otherwise we do nothing.

Figure 5: A portion of the lattice with the allowed moves and the respective rates. Beads are drawn as black lozenges. When p=1,q=0 or p=0,q=1 the process can be seen as an infinite set of discrete Hammersley processes, one per column, each interacting with the two neighboring processes. If instead we allowed particles to jump only by \pm1 with rates p/q, the process would be the single-flip dynamics described in the introduction and would correspond to an infinite collection of mutually interacting asymmetric simple exclusion processes (ASEP), one per column.
Figure 5: A portion of the lattice with the allowed moves and the respective rates. Beads are drawn as black lozenges. When or the process can be seen as an infinite set of discrete Hammersley processes, one per column, each interacting with the two neighboring processes. If instead we allowed particles to jump only by with rates , the process would be the single-flip dynamics described in the introduction and would correspond to an infinite collection of mutually interacting asymmetric simple exclusion processes (ASEP), one per column.

It is not obvious that the process is well defined on the infinite lattice. The danger is that beads could escape to or to in finite time (even in an arbitrarily small time). This may occur when spacings between beads in the initial configuration grow sufficiently fast at infinity. The problem is that the rate at which a bead moves, say, upward is and the average size of the jump is , and is not bounded.

Our first result (Theorem ?) is that the process is well defined for almost every initial condition sampled from and that is invariant. “Well-defined” means that the displacement of every bead with respect to its position at time zero is almost surely finite for every . In the symmetric case , assuming that the process is well defined, invariance of the Gibbs measure is obvious because it is reversible.

To precisely formulate the result, let us start by defining, given , a cut-off process where -clocks at distance more than (resp. -clocks at distance more than ) from the origin of are switched off. As long as there is no problem in defining the process on the whole , since this is effectively a Markov jump process on a finite state space (once a particle is inside the ball of radius it cannot leave it and therefore there is only a finite number of particles, determined by the initial condition, that can ever move). We call the configuration at time , started from initial condition . Given a column , let be the position of its bead at time , with . The label is assigned in the initial condition and is attached to beads forever. For instance, one can assign the label to the lowest bead in with non-negative vertical coordinate (in the initial condition). We assume hereafter that in each column there is a doubly infinite set of beads, i.e. the index runs over all of .

Two processes with different cut-offs and can be coupled in the obvious way: their -clocks (resp. -clocks) are the same in the ball of radius (resp. ). It is then easy to check that is increasing w.r.t. and decreasing w.r.t. . We will then define

to be the position of the -th bead at time for the process without cut-off.

Assuming that is finite for every , call the corresponding bead configuration and let be the law of the process started with initial distribution (if is concentrated at some , then we write just ).

Here, a function is said to be local if it depends only on for some finite . It is also possible to see (cf. Remark ?) that the limit does not depend on the order how one takes the limits and .

Theorem ? is proven partly in Section 6 (existence of the dynamics) and partly in Section 8.1 (invariance of ).

It is a relatively standard fact to deduce from Theorem ? that, if we start from conditioned to have a bead say at the origin, then the law of the dimer configuration re-centered at the time-evolving position of this marked bead (tagged particle) is time-independent, see Section 8.2. More precisely, fix a horizontal edge of . Given an initial condition such that there is a bead at , call its vertical coordinate at time (the horizontal coordinate does not change). Let also , with the vertical translation by , be the dimer configuration viewed from the tagged bead and call the law of the process started from some initial distribution . Finally, let be the Gibbs measure conditioned on the event that there is a bead at .

3Interface speed and fluctuations

The stationary states are characterized by an upward or downward flux of beads, according to whether or . The particle flux is directly related to the average height increase in the stationary state. While the height function was defined only up to an additive constant, one can define unambiguously the increase of the height at a face from time to : equals the number of beads that cross the face downward up to time , minus the number of beads that cross it upward.

For each horizontal bond let (resp. ) be the lowest (resp. highest) bead in the column of , at vertical position strictly higher (resp. strictly lower) than . Also, call the collection of hexagons that has to cross to reach position and set if this move is not possible (keeping the other beads where they are). Define similarly.

The following result identifies the average height drift and shows that the fluctuations of in the stationary measure are smaller than any power of :

Note that only edges in the same column as and above it can contribute to . The value of is independent of by translation invariance of . The r.h.s. of is linear in because of stationarity of and linear in because the stationary state does not depend on .

Theorem ? is proven in Section 9.

It is not obvious to compute explicitly in terms of the slope , starting directly from the determinantal representation of the Gibbs states. In [4], A. Borodin and P. L. Ferrari considered the dynamics for for a special, “integrable”, initial condition , whose height function is deterministic and has non-constant slope (see Fig. 1.2 of [4]: lozenges with a dot correspond to our south-east oriented lozenges, white squares to our north-east lozenges, while dark lozenges correspond to our beads). Let us emphasize that with such initial condition, each bead has a deterministic lowest position it can possibly reach on its column (this is related to the fact that in [4] there is no dotted lozenge with coordinate ), so that the well-posedness of the process poses no problem in that case. One of the results of [4] is a hydrodynamic limit, that in our notations we can formulate as follows: for every and one has

and satisfies

(this corresponds to formulas (1.9)-(1.11) in [4], after after a suitable change of coordinates due to the fact that in [4] the height is not taken with respect to the horizontal plane and a different reference frame than our frame is used). From this, one can naturally guess that in should be given by

Since and are in , the above expression is immediately seen to be positive. After a first version of this work was completed, Chhita and Ferrari [9] proved, through a smart combinatorial identity based on the determinantal structure of the Gibbs states, that indeed holds.

By the way, Proposition 3.2 of [4] says that the law of local dimer observables around point at time tends as to that of the same observables under the Gibbs state of slope . On the basis of this, it was natural to conjecture that our Theorem ? holds.

Referring to , we believe that the order of magnitude of the variance of is actually : this is indeed the result found by Borodin and Ferrari [4], in the particular case where , and for the special initial condition mentioned above. In this respect, our method allows indeed to refine estimate , under a (purely technical, we believe) condition on the slope , to the following:

For instance, if (the three types of dimers have density , in which case are all equal) one finds, evaluating numerically the integral in , that the l.h.s. of is , so that holds. By continuity, this remains true in a whole neighborhood of while, again numerically, does not seem to be satisfied in the whole set of non-extremal slopes .

Let us stress once more that we believe to hold for every non-extremal and to be of the optimal order w.r.t. , while we do not attach any particular meaning to condition .

3.1Extension to dominos (perfect matchings of )

Our result extends to perfect matchings of , or equivalently domino tilings of the plane, cf. Figure 6: also in this case, one can define an asymmetric Markov dynamics (the height function has a non-zero drift) that leaves the Gibbs states invariant. We give only a sketchy description of the generalization, omitting those details that are identical to the case of the honeycomb lattice.

Figure 6: The correspondence between dimer covering (perfect matching) of \mathbb Z^2 and domino tiling of the plane.
Figure 6: The correspondence between dimer covering (perfect matching) of and domino tiling of the plane.

Since is bipartite we can color its vertices black/white with the rule that each vertex has neighbors only of the opposite color. The height function on the set of faces of can be defined (modulo an arbitrary additive constant) as follows: for each choose any nearest-neighbor path from to and set

with the sum running over the edges crossed by the path, according to whether crosses with the white vertex on the left/right and the indicator function that there is a dimer at . The definition is independent of the choice of path.

The classification of translation-invariant ergodic Gibbs states is analogue to the honeycomb lattice case (actually the structure is the same for all planar, periodic, infinite bipartite graphs [22]): there exists an open polygon (for the lattice it is a square, while for it is a triangle, as discussed in Section 2.2) such that for every (non-extremal slope) there exists a unique translation-invariant ergodic Gibbs state satisfying

where the vectors are as in Figure 7. The determinantal representation still holds, with a different polynomial that however still has two simple zeros on the torus . In order to define the irreversible dynamics that leaves the Gibbs states invariant, we have to find an analogue of the “columns” and “beads”. This is inspired by [24]. The set of square faces of is sub-divided into infinite “columns” (indexed by ), i.e. diagonally oriented zig-zag paths, see Figure 7. Dimers that occupy an edge across a column are called “beads”. Each column is oriented along the positive direction, so it makes sense to say that a bead in column is above a bead in the same column.

Figure 7: A portion of \mathbb Z^2 with vertices colored black/white. The colored graph is periodic, i.e. invariant w.r.t. translations by n_1\vec e_1+n_2\vec e_2, n_1,n_2\in\mathbb
  N. The white and shaded zig-zag regions are the columns, indexed by \ell. Columns are oriented in the positive \vec e_1 direction. The two thick bonds, transversal to columns, represent two beads.
Figure 7: A portion of with vertices colored black/white. The colored graph is periodic, i.e. invariant w.r.t. translations by . The white and shaded zig-zag regions are the “columns”, indexed by . Columns are oriented in the positive direction. The two thick bonds, transversal to columns, represent two beads.

Given columns , call the set of vertices of shared by the two columns and order the sites of according their coordinate. Then, a bead on column is said to be higher than a bead on if the vertex of on is higher than the vertex of on . With this definition, it is easy to see that beads satisfy the same interlacement property as on the honeycomb graph: given beads on , there exists on and on with and . Also, like on the honeycomb lattice, it is easy to see that if there is at least a bead in each column, then it is possible to reconstruct the whole dimer covering knowing only the bead positions.

The dynamics is then defined as follows. Assign to any possible bead position, i.e. to each edge that is transversal to some column, two independent Poisson clocks of rates and , as before. All clocks are independent. When a -clock (resp. -clock) at edge of column rings, if there is a bead at then do nothing. Otherwise, move the first bead below (resp. above) in column to position , provided this does not violate the interlacing constraints. Note that the dynamics is the same as on the honeycomb lattice, only the definition of “column” and “bead” being lattice-dependent. Observe also that each move can be seen as a concatenation of elementary moves on adjacent faces along the same column, each elementary move consisting in the rotation by of two dimers on the same face of (Figure 8).

Figure 8: The two elementary moves of the domino dynamics
Figure 8: The two elementary moves of the domino dynamics

In fact, the effect of an elementary move is to shift a single bead one position up or down along its column. Note that, like in the case of the hexagonal lattice, when a bead moves one step upward crossing a face , the height function at changes by .

As in Section 3, given an edge transversal to some column , call the highest bead in , strictly lower than and let the collection of square faces of that crosses when it is moved to (with if the move is not allowed). Then:

With the exception of Section 4.2, in the rest of the work we will always consider the case of the hexagonal lattice.

4Dynamics on the torus

4.1Honeycomb lattice

We will let the torus denote the hexagonal graph , periodized (with period ) along directions and we assume that . Note that now columns along which beads move are “circles” containing hexagonal faces. We will say as before that a bead moves “upward” or “downward”, but what we mean is that it moves in the positive or negative direction around the torus.

Let be the set of configurations such that the height changes by (resp. ) along any closed path winding once in the positive (resp. ) direction. On each column there are beads and bead positions on neighboring columns are again interlaced. We denote the uniform measure over . It is known that converges weakly to , if the configuration space is equipped with the product topology [22]. Essentially, averages of bounded local functions converge.

On the process is defined similarly as in Section Section 2.3 for the infinite graph. For instance, when a -clock at an edge rings, one moves to the first bead that is found when proceeding in the direction from along the same column, unless this move is forbidden by the interlacing constraint. The process is ergodic on , actually it is known that we can go from any configuration to any other by positive-rate elementary moves as in Figure 4 (see [11] for details).

It is actually easy to deduce, using ergodicity of the process in each of the sectors , that the only stationary measures are convex combinations of .

Call the generator of the process. We want to check that

(stationarity of ) . One can decompose the generator as with involving only the up-jumps (related to the -clocks) and the down-jumps. It is sufficient to prove that , for the argument being the same. For every we have . Given let be the collection of that can be reached from by a single non-zero up-jump (not necessarily of length one) of a bead and let be the collection of from which one can reach with a single non-zero up-jump of a bead. For every we have , while simply because the sum of row elements of the generator is zero. We see then

We want to see that . Note that while , with the sum running over beads and being as in Definition ?2. We will prove that is independent of : as a consequence, it must be zero because the sum over of is zero. Assume that differs from only by a single elementary up-move of some bead on some column . Then, after the move the only beads that may have changed their values of are itself and , with the bead in column that is “just above

Figure 9: The configurations \sigma and \sigma' around b. In this example, once the bead b moves up by 1, I^-_{b^+_{\ell-1}} decreases by 1, I^+_{b^-_{\ell+1}} increases by 1 while I^-_{b^+_{\ell+1}} and I^+_{b^-_{\ell-1}} stay constant.
Figure 9: The configurations and around . In this example, once the bead moves up by , decreases by , increases by while and stay constant.

(see Figure 9) and analogously for the others. It is clear that the contribution of to is : indeed, decreases by and increases by . Then look at column . One of the following two mutually exclusive cases occurs (Figure 9): either increases by and stays constant or stays constant and decreases by . In both cases, the net variation of from column is . The same holds for column (since we are assuming , columns are distinct). Altogether, . We have proved that is unchanged if we perform an elementary up-move. Given that the space state is connected, we proved that is constant (and therefore zero) on .

The analog of Proposition ? for the dynamics on the torus is the following.

The proof is very similar to that of Proposition ?. Call the part of the generator of the process involving only -clocks. We have to show for every

A symmetric argument then gives .

The measure is uniform among the configurations with a bead at . We have equal times the number of configurations different from that can be reached from with a single move. The configuration can change either because a bead different from (the bead that is at ) moves, or because itself moves and then the dimer configuration has to be re-centered around the new tagged particle position. Note indeed that, when moves, necessarily the configuration viewed from it changes, since the distance from the first bead above it decreases. The number of reachable configurations is then . Similarly, one sees that

Then, the l.h.s. of equals the r.h.s. of (only with replaced by ), that we know to be zero.

A “gradient condition”

The bead dynamics on the torus has an trivial conserved quantity: the number of particles. There is however a less obvious one. For each of the columns define

with the sum running over the beads of column . We have seen in the proof of Proposition ? that the “total charge” is exactly zero. A simple computation shows that, when , the instantaneous drift of is


This is a “gradient condition” [30]: the derivative of the charge at is given by the divergence of a current, here , which is itself the gradient of a function of the configuration.

As we mentioned in the introduction, conditions of this type are typically the key to guarantee that a reversible Gibbs measure remains invariant once an external driving field that breaks reversibility is introduced, see e.g. [2] and [20]. The unusual fact here (with respect to the more standard framework of e.g. the simple exclusion or zero range processes) is that the current associated to the local charge does not seem to satisfy a gradient condition, while that of the non-local charge (integrated along the columns) does. Note that on the infinite lattice is not well-defined (it is just infinite).

4.2Square lattice

The finite graph with periodic boundary conditions is defined like for the honeycomb lattice, except that the directions along which one periodizes are now , see Figure 7. Note that each periodized column is a “circle” containing square faces. The measure is defined as the uniform measure over dimer coverings of such that the height changes by when winding once in the direction, and tends to as for every local observable [22].

Like for the honeycomb lattice, one has

The only point where the proof differs w.r.t. the honeycomb lattice case is the way one shows that , as after . Recall that it is sufficient to show that, after any elementary move, the difference is unchanged, whatever the initial configuration is.

When an elementary move is performed at a face in column , a bead jumps from an edge to that has a common vertex with . This common vertex belongs to either or (recall that is the set of vertices common to columns ). Assume w.l.o.g. that the former is the case, as in Figure 10, and that is higher than in column . After the move, decreases by and increases by . On the other hand, it is clear that is unchanged for beads on column , or on any other column except . Therefore, we have to find a change of coming from column . Call , resp. , the first bead above (resp. below) in column , and call the bead “between” and in column . (The notion of ordering for beads in neighboring columns was introduced in Section 3.1). Then, with reference to Figure 10, note that:

Figure 10
Figure 10
  • if is at or higher than edge , then is at or higher than and is the same, irrespectively of whether is at or . On the other hand, edges are accessible to if is at and are not if is at , so differs by in the two cases. Altogether, when is moved from to , the contribution of to the change of is , as desired;

  • symmetrically, when is at or lower than then is at or lower than . When is moved from to , does not change, while decreases by , since are not available positions any more. Again, we get a change for , this time coming from .

  • finally, suppose that is at or . If is at then position is available for and is not available for , while if is at the opposite holds. As a consequence, both and contribute to the change of .

Deducing stationarity of on the infinite graph from stationarity of on the torus works exactly the same on or ; for definiteness, in Section 8 we will stick to the former case.

5The discrete Hammersley dynamics (DHD)

On the way towards Theorem ?, let us switch for a moment to a one-dimensional interacting particle system known as Discrete Hammersley Dynamics (DHD) [14]. The configuration space of the DHD consists of particle configurations on (at most one particle per site). Each site of has an i.i.d. Poisson clock of rate . When a clock rings at a site , if the site is occupied then nothing happens; otherwise, take the first particle to the right of and move it to . Note that each particle moves to the left with rate equal to the number of empty sites before the next particle to the left, and the new position is uniform among the sites. We call the position of the particle () at time . Particles are labelled in the initial condition in such a way that , with some arbitrary choice of whom to label (for instance, it could be the first particle to the right of the origin). Labels do not change as particles move.

The works [1] consider instead the (continuous) Hammersley process [1], which is defined similarly as the DHD, except that particles live on instead of : again, each particle moves to the left with rate equal to the available space before the next particle and the new position is chosen uniformly in the available interval. In [28] it is proven (among many other results):

Theorem ? extends immediately to the DHD [14] and is obtained with the help of a Harris-type graphical construction, that we recall here. To each site of associate an independent Poisson point process of density on : this is the set of times when the clock at that site rings. Given a realization of all these i.i.d. Poisson processes and given , , we can consider the set of all possible up-right paths in the rectangle , i.e. sequences of space-time points in the point process in the rectangle, with and . Note that inequalities are strict (for times this is not restrictive since with probability one there is at most one clock ringing at a given time). Let as in [1] be the maximal number of points of the Poisson processes on one such path. Let also

Then (this is given in [1] in the continuous Hammersley process where and are defined similarly, but the same holds true also for the DHD) for every

Note that the DHD has the following monotonicity property:

This is immediate from : if some is changed to , then increases at most by .

The representation also allows to get an upper bound on the probability that the displacement of a particle is large. Indeed, if then there exists such that

With a union bound, the probability (conditionally on the initial positions) that is upper bounded by

where denotes the expectation only with respect to the Poisson clocks. One has3

Indeed, there are strictly increasing distinct sequences . Given one of these, the probability that there is an up-right path equals the probability that a Poisson random variable of average equals at least . On the other hand, if is a Poisson variable of average then for

because . Then, follows from

Let us call the law of the DHD started from an initial configuration . From we have then