Speed and fluctuations for some driven dimer models
We consider driven dimer models on the square and honeycomb graphs, starting from a stationary Gibbs measure. Each model can be thought of as a two dimensional stochastic growth model of an interface, belonging to the anisotropic KPZ universality class. We use a combinatorial approach to determine the speed of growth and show logarithmic growth in time of the variance of the height function fluctuations.
We consider two-dimensional stochastic growth models in the anisotropic KPZ universality class . Stochastic interface growth models have a random local growth mechanism which is (effectively) local in space and time, , but with smoothing mechanisms that ensure deterministic growth under hydrodynamic scalings. In two dimensions, the average speed of growth of the interface in a stationary state can be parameterized by the slope of the height function. The anisotropic KPZ universality class contains the models for which the signature of the Hessian of the speed of growth is . This is in contrast with the usual, isotropic, KPZ universality class where the signature is or . In the anisotropic case, it is expected that the fluctuations of the height function behave asymptotically like as grows . This has been analytically verified for some exactly solvable models [21, 4, 3] and confirmed by numerical studies [18, 11]. Furthermore, it is expected that on large space-time scales and modulo a linear transformation of space and time coordinates, the height function fluctuations of the stationary process have the same asymptotic correlations as those found in the stochastic heat equation with additive noise (see [2, 1] for recent works).
In this paper we consider two dimer models on infinite bipartite graphs and (the honeycomb graph). Dimers, that are viewed as particles, perform long-range jumps with asymmetric rates. For the honeycomb graph, the dynamics were defined in  and later extended to a partially asymmetric situation in . The dynamics on was introduced in . For both these models, translation-invariant stationary measures for interface gradients are Gibbs measures on dimer configurations with prescribed dimer densities [5, 17, 16]. In , the specific prescribed initial conditions were not stationary but this choice had the useful property that in a large enough subset of space-time, dimer correlation functions were determinantal. This allowed, among others, the computation of the law of large numbers and to determine that the variance of the height function behaves asymptotically like and has Gaussian fluctuations on that scale. However, this determinantal property for space-time correlations, which allowed for explicit computations, is no longer true for the partially asymmetric dynamics or for those with stationary initial conditions.
In this paper we consider stationary initial conditions and obtain two results, that apply equally to the totally asymmetric or to the partially asymmetric situation. The first one is the speed of growth for the model on (Theorem 2.3). The difficulty here is to find a compact and explicit formula for the speed of growth, since by definition of the dynamics, is given by an infinite sum of probabilities of certain dimer configurations and therefore by an infinite sum of determinants involving the inverse Kasteleyn matrix. To obtain this result we mimic the approach used for the honeycomb lattice in . There, a combinatorial argument showed that the infinite sum reduces to a single entry of the inverse Kasteleyn matrix, leading to the explicit formula (2.5). For , this is no longer the case, but we are able to prove that the infinite sum is given in terms of a few explicit entries of the inverse Kasteleyn matrix. As a side result, we verify explicitly that the signature of the Hessian of is .
The second result concerns the logarithmic growth of variance of the height function for the honeycomb graph, see Theorem 2.4 (the method can be extended to the dynamics on but in order not to overload this work we skip this). This result was partially proved in , with a technical restriction on slope . Our new approach simplifies the proof contained in  and it extends its domain of validity to the full set of allowed slopes.
The rest of the paper is organized as follows. In Section 2 we define the models and give the results. Section 3 contains the background on dimers models. Theorem 2.3 on the speed of growth on is proved in Section 4. Theorem 2.4 on the variance is proved in Section 5.
F. T. was partially funded by the ANR-15-CE40-0020-03 Grant LSD, by the CNRS PICS grant “Interfaces aléatoires discrètes et dynamiques de Glauber” and by MIT-France Seed Fund “Two-dimensional Interface Growth and Anisotropic KPZ Equation”. P.F. was supported by the German Research Foundation as part of the SFB 1060–B04 project.
2 The growth models and the results
2.1 Perfect matchings and height function
We are interested in two infinite, bipartite planar graphs in this work: the grid and the honeycomb lattice . In both cases, we let denote the set of perfect matchings or dimer coverings of , i.e., subsets of edges in (dimers) such that each vertex is incident to exactly one edge. Both graphs are bipartite, so we can fix a -coloring (say, black and white) of their vertices , see Figures 1 and 2. We denote (resp. ) to be the set of white (resp. black) vertices of .
Associated to each dimer covering , there is a height function defined on faces of , as follows: is fixed to zero at some given face of (the “origin”) and its gradient are given by
where: are faces of , is any nearest-neighbor path from to (the r.h.s. of (2.1) does not depend on the choice of ), the sum runs over edges crossed by , equals (resp. ) if is crossed with the white vertex on the right (resp. left) and is a function defined on the edges of , such that for any ,
where means that is incident to . A standard choice for the square lattice is ; for the hexagonal lattice, we let if is horizontal and otherwise.
As we recall in more detail in Section 3.1 below, for both graphs there exists an open polygon such that for every there exists a unique translation invariant and ergodic Gibbs probability measure on dimer coverings of , denoted . With the choice of coordinates we make in this work (see Section 3.1), the polygons for the two graphs are as follows:
is the open triangle in with vertices , and is the open square in with vertices .
2.2 Particles and interlacement conditions
A common feature of the two graphs and , that makes them special with respect to other planar, bipartite graphs, is that to any one can associate a collection of “interlaced particles”. First of all, we partition the set of faces of into disjoint “columns” . In the case of , a column consists in the set of faces with the same horizontal coordinate, while for it is a zig-zag path as depicted in Fig 2.
We call the set of vertices of shared by and . Vertices can be ordered in a natural way and we will say that if precedes in the upward direction (for ) or in the up-left direction of Figure 2 (for ). An edge of will be called “transversal” if it has one endpoint on and the other on for some . Dimers on transversal edges will be called “particles”.
Given two particles and , each with one endpoint (say respectively) on the same , let us say that is higher than (we write ) if . The following interlacement condition is easily verified both for and : given two particles on the same column and verifying , there exists a particle on and a particle on such that , . See Figures 1 and 2.
Both on and on it is easy to check that, under the assumption that every contains at least one particle, the whole dimer configuration is uniquely determined by the particle configurations. In the situation we are interested in, there are almost surely infinitely many particles on each ; therefore, we will implicitly identify a dimer configuration and the corresponding particle configuration.
2.3 Dynamics and new results
We describe here the growth dynamics of  in a unified way for and . We need some preliminary notation. Given a transversal edge on column , let denote the highest particle in column that is strictly below . Given a configuration , we say that “particle can reach edge ” if the configuration obtained by moving to edge while all other particles positions are unchanged still satisfies the particle interlacement constraints, i.e., .
The continuous time Markov chain of , in its totally asymmetric version, can be informally described as follows. To each transversal edge of is associated an i.i.d. exponential clock of mean . When the clock at rings, if particle can reach without violating the interlacement constraints then it is moved there. If cannot reach , then nothing happens.
Note that the size of particle jumps are unbounded, so it is not a-priori obvious that the definition of the Markov process is well-posed. However, one of the results of  is that given any , for almost every initial condition sampled from the Gibbs measure the dynamics is well-defined (i.e., almost surely no particle travels an infinite distance in finite time). Also, it is proved there that the measures are stationary for the dynamics. We let denote the law of the stationary process started from .
Note that when is moved from its current position in column say to the edge in the same column, it jumps over a certain number of faces of . We define the “integrated current” as the total number of particles that jump across a given face of the graph, say across the face that was chosen as origin, from time to time ( is trivially related to the height change at ). In  it was proven:
For every , there exists such that
Moreover, if then there exists a non-empty subset of such that, for every ,
Later, in , the function for was computed explicitly111In this work we use different conventions as in  for lattice coordinates and this is the reason why formula (2.5) looks different from formula (3.6) of :
Our main results here complete the above picture as follows:
For the dynamics on , the speed of growth is given by
where for takes value in .
For , (2.4) holds for every .
Moreover, the proof of (2.4) we give here is substantially simplified w.r.t. the one in . Also, our method can be easily adapted to prove Theorem 2.4 also for the dynamics on and every but, in order to keep this work within a reasonable length, we do not give details on this extension.
The work  studies a more general, partially asymmetric dynamics where upward jumps have rate and downward jumps have rate . In this case, the speed of growth is given by the above formulas multiplied by . Also, the result on the variance holds true also for the partially asymmetric version. In fact, from [23, Sec. 9] one sees that Theorem 2.4 holds for general as soon as Theorem 3.1 below, that is independent of , is proved.
2.4 Geometric interpretation of
The stationary and translation invariant Gibbs measures form a two-parameter family. From an interface perspective, it is natural to use the average slope of the interface, , as parametrization. Then all other quantities, such as the average number of dimers of a given type or the speed of growth, are functions of . As it was already known for the honeycomb lattice, the correlation kernel giving dimer correlations, that in principle is a double contour integral , can be rewritten as a single integral from to , where is a complex number in the upper half plane . Further, for and for a special initial condition, it was shown  that the height field fluctuations of the growth model converges to a Gaussian free field (GFF). More precisely, the correlations on a macroscopic scale at different points converge to the correlations of the GFF on between the points obtained by mapping the points to by . The map was known already from the work of Kenyon  (there it is called in Section 1.2.3 and Figure 2). A generalization of  to a setting with two different jump rates was made in .
Here we shortly present how the densities of the different types of dimers, the correlation kernel and the speed of growth are written in terms of . For the hexagonal lattice we refer to : the three types of dimers are in Figure 5.1, the points , and form a triangle whose internal angles are times the frequencies of the types of dimers (Figure 3.1), and the correlation kernel as a single integral is given in [4, Prop. 3.2]. Finally, an interesting property is that the speed of growth (2.5) equals .
For the square lattice, there also exists (not the same one as for the hexagonal lattice) such that the correlation kernel is given as a single integral from to (see Lemma A.1). Using this and formula (3.2) below, one can easily compute the densities of the different types of dominoes with the result
which, remarkably, is the same form as in the hexagonal case. As in the hexagonal case, also in the square case has a nice geometric representation in terms of dimer densities, see Figure 3.
3.1 Gibbs Measures
An ergodic Gibbs measure or simply a Gibbs measure , in our context, is a probability measure on that is invariant and ergodic w.r.t. translations in and satisfies the following form of DLR (Dobrushin-Lanford-Ruelle) equations: for any finite subset of edges , the law conditioned on the dimer configuration on edges not in is the uniform measure on the finitely many dimer configurations on that are compatible with . By translation invariance, to a Gibbs measure one can associate an average slope , such that
with the coordinate unit vectors.
It is convenient, both for this section and the rest of the work, to make an explicit choice of coordinates on . Let us start with the graph . Both white and black vertices are assigned coordinates . The two (white and black) endpoints of the same north-west oriented edge will be assigned the same coordinates (we will denote them ) and we make an arbitrary choice of which edge has endpoints of coordinates . The coordinate vectors are chosen to be the unit vectors forming an angle and , respectively, w.r.t. the horizontal axis. See Figure 1. Note that the nearest neighbors of the black vertex are the white vertices and .
As for , we let be the vectors forming an angle and w.r.t. the horizontal axis, see Figure 2. Again we fix arbitrarily the origin of the lattice and we establish that a white vertex has the same coordinates as the black vertex just to its left. The nearest neighbors of the black vertex are the white vertices .
Recalling the definition of height function it is easy to see that, for any Gibbs measure , the slope must belong to the closure of the polygon of Definition 2.1.
It is known  that for every there exists a unique Gibbs measure with slope . This can be obtained as the limit (as ) of the uniform measure on the subset of dimer coverings of the periodization of the lattice such that the height function changes by along a cycle in direction .
The correlations of the measure have a determinantal representation , that we briefly recall here. First of all, one needs to introduce the Kasteleyn matrix: this is the infinite, translation invariant, matrix , with rows/columns indexed by black/white vertices of . Matrix elements are non-zero complex numbers for nearest neighbors and are zero otherwise. The non-zero elements depend also on the slope . See below for the explicit expression of for the graphs and . Next, one introduces an infinite, translation-invariant matrix (as the notation suggests, equals the identity matrix). Again, see below for the expression of for and . All multi-point correlations of can be expressed via and as follows : given edges ,
The definition of matrices is not unique and different choices than the one we make below can be found in the literature.
For , we let
where are such that in the triangle with sides , the angle opposite to the side of length is , with , and . Note that (resp. ) is the density of dimers oriented horizontally (resp. oriented north-west, north-east). The inverse Kasteleyn matrix is
with and and the integral runs over the anticlockwise circles in the complex plane.
For we take instead
(the “magnetic fields” are fixed by the slope as specified below) and the inverse Kasteleyn matrix is given by
where the integral runs over and
The parameters are related to the slope as follows:
This is simply because, by the definition of height function, one has for instance
Injectivity of the map is related to the fact that is the gradient w.r.t. of a surface tension function that is a convex function of .
3.2 Average and variance of the current
For ease of notation, given distinct edges of , we let
where we recall that denotes the dimer covering.
3.2.1 Average current
Let denote the index such that the face of that we established to be the origin is in column and let denote the set of edges , transversal to , that are above . Then, from the definition of the dynamics, we obtain the following expression for the speed :
simply because, if can reach , it will do so with rate and with such an update, it will increase the integrated current through by .
Then, (3.13) can be expressed more explicitly. In view of Theorem 2.3, we consider the case . With reference to Figure 4, where for convenience we rotated the graph by clockwise, we first notice that consists of the set of transversal edges .
Also, it is easily checked that the event is equivalent to the event that edge is occupied by a dimer while is not. Finally, the event , , is equivalent to the event that edges are all occupied by dimers.
As a consequence,
In Section 4 we will show that the r.h.s. of (3.14) equals the r.h.s. of (2.6). The sum is convergent: in fact, label the faces in the column , where is adjacent to and above . Then, the event is equivalent to . On the other hand, so that
3.2.2 Variance of the current
Let us move to the variance of for (we do not work out formulas for ). Recall that, given a dimer configuration and a horizontal edge , we denote the highest particle below in the same column. We denote the number hexagonal faces that has to cross in order to reach edge and we set if cannot be moved to (i.e., if the move violates interlacements).
Denote by the set of horizontal edges with . Then, for every there exists a constant such that
3.3 Dimer coverings of bipartite graphs
In the following, we will need more general bipartite graphs than just and . To each of the edges , we assign a positive number called an edge weight. We denote the weight of the edge by with . We denote the set of dimer coverings by and, if the graph is finite, we denote the partition function by . That is,
We define to be the dimer model probability measure on the graph , that is for , .
Given a subset of edges and a subset of vertices , we write to be the graph with all the edges in and vertices in removed from , along with the edges incident to either or . Let denote the partition function of this graph (if either or is empty we omit it from the notation).
We use to denote the Kasteleyn matrix of which has columns indexed by the black vertices and rows indexed by white vertices with entries given by
where is a modulus-one complex number chosen so as to satisfy the following property. Given a face of the graph, let be the edges incident to it, ordered say clockwise with an arbitrary choice for . Then, we impose that
This is called a Kasteleyn orientation. Existence of a Kasteleyn orientation for every (bipartite) planar graph is known  and in general many choices are possible. When is a bipartite sub-graph of the infinite lattice or and , the restriction of to does not in general provide a correct Kasteleyn orientation for and this will be an important point later.
Kasteleyn [12, 13] and independently Temperley and Fisher  noticed that for domino tilings (to be more precise, their formulations involved the more complicated non-bipartite graphs but the above formulation is sufficient for this paper). This identity is true irrespective of the choice of Kasteleyn orientation and holds for any bipartite finite planar graph. An observation due to Kenyon  shows that statistical properties can be found using the inverse of the Kasteleyn matrix, that is, for edges in the graph ,
Given an edge weight function , define face weights as the alternating product of the edge weights: given a face of adjacent to edges (say in clockwise order with a given choice of ), let
The dimer model probability measure is uniquely parametrized by its face weights, which means that two edge weight functions lead to the same probability measure if the corresponding face weights are equal. Suppose that and are two such edge weight functions. Then there exist functions and on white and black vertices respectively such that for each edge . We say that and are gauge equivalent and the act of multiplying edge weights by functions defined on its incident vertices is called a gauge transformation. The Kasteleyn matrix for a gauge equivalent weighting is obtained by pre-and post-composing with diagonal matrices built from the gauge transformation functions.
4 Speed of growth on
In this section, we prove (2.6). We begin by remarking that the edges that appear in formula (3.14) are the edges while are the edges ; see the left picture in Figure 4. We also remark that and are the edges and for respectively; see the right picture in Figure 4. Set and with the convention that .
4.1 Finite Graph
Consider a finite bipartite graph contained in and set all edge weights to . Throughout this section, we denote to be the vertex and to be the vertex . See Figure 5.
With the notation of Definition 3.2 we have
and we assume that is large enough to include and .
The above lemma and its proof have a similar flavour to [6, Proposition 3.5] with the key difference that and are not on the same face.
Consider the graph . There are three possibilities for the dimers incident to the vertex . These are given by the edges , and ; see Figure 6.
If a dimer covers the edge , then the remaining graph is the same as . If a dimer covers the edge instead, the remaining graph is the same as . This gives (remember that all edge weights equal )
which can readily be seen from Figure 6.
For , we have inductively the equations
Indeed, (4.4) follows because from the graph , there are two possible dimers covering the vertex : either the edge is covered by a dimer or the edge is covered by a dimer; see Figure 7 for the case when . Then, (4.4) follows after noticing that the graph is the same as the graph .
Similarly, to show (4.5), there are two possible dimers covering the vertex which are or . Then (4.5) follows after noticing that the graph is the same as the graph . We substitute the recursions in (4.4) and (4.5) into (4.3) to give
We divide the above equation by and use the fact that . The claim is proved. ∎
Recall that denotes the Kasteleyn matrix of . If we want a Kasteleyn matrix for , we cannot just take the restriction of . The problem is that the, since for the four square faces of around both and (recall (3.19) for the definition of ), for the square faces that has around we get which does not satisfy (3.19). This is easily fixed: to define a valid Kasteleyn orientation on we need to reverse the orientation of a ‘path of edges’ connecting the two faces . In our case, as we explain below, it is sufficient to reverse the orientation of a single edge. A similar idea on a much more complicated scale was used in great success in  to find correlations in the monomer-dimer model.
We set to be the vertex , to be the vertex , to be the vertex and to be the vertex ; see Figure 5. For simplicity, we organize the matrix so that and are in columns and while , , and are in rows to (in that order).
We let be the matrix obtained from the matrix by removing the rows and columns associated to the black and white vertices from respectively, for some collection of vertices .
Further, define for and by
Observe that is a Kasteleyn matrix for , that satisfies (3.19).
The following lemma relates with entries of the inverse of .