Conformal invariance of isoradial dimers
An isoradial graph is a planar graph in which each face is inscribable into a circle of common radius. We study the 2-dimensional perfect matchings on a bipartite isoradial graph, obtained from the union of an isoradial graph and its interior dual graph. Using the isoradial graph to approximate a simply-connected domain bounded by a simple closed curve, by letting the mesh size go to zero, we prove that in the scaling limit, the distribution of height is conformally invariant and converges to a Gaussian free field.
One of the most important starting points for two-dimensional critical lattice models in statistical physics is the assumption that the continuous limit is universal and conformally invariant. It says that, in the limit when the lattice spacing goes to zero, macroscopic quantities of the model transform covariantly under conformal maps of the domain (conformal invariance) and are independent of the lattice (universality). Under the assumption of universality and conformal invariance, physicists have successfully predicted exact values of certain critical exponents. However, the conformal invariance and universality assumptions were beyond the mathematical justification until very recently ([22, 17, 20, 3, 11]) .
In this paper, we focus on the perfect matching on planar graphs. A perfect matching, or dimer covering of a finite graph is a set of edges covering all the vertices exactly once. The dimer model is the study of random dimer coverings of a graph. By assigning weights to edges, one can define a probability measure on random dimer coverings (). The height function is a random function which assigns a unique number to each faces of the graph for each realization of random perfect matchings. If the underlying graph is a subgraph of the square grid , Kenyon ([13, 14]) proved the conformal invariance of height distribution in the scaling limit under certain boundary conditions, given the uniform measure of random perfect matchings.
An isoradial graph is a planar graph in which each face is inscribable in a circle of common radius. It was introduced by Duffin () in the late sixties, in an equivalent form of rhombic lattices, and reappeared recently in the work [19, 15, 2, 3, 1, 5, 9]. The isoradial graph might be the largest class of graphs where the complex analysis techniques have a “nice” discrete analog, and hence a natural setting for the universality assumption, which includes, and is more general than the class of regular graphs, see  for an exposition of the discrete complex analysis technique on isoradial graphs. Ising models on isoradial graphs have been studied extensively in [19, 1, 3], and spectacular results were proved including conformal invariance and universality (). For perfect matchings on isoradial graphs, Kenyon () proved an explicit form for the operator of bipartite isoradial graphs on the whole plane. Following that, it is proved in , that the height function of perfect matchings on the whole plane converges to a Gaussian free field, yet the conformal invariance and boundary conditions were not addressed in  since the paper deal with graphs on the whole plane.
In this paper, we use isoradial graphs to approximate an arbitrary simply connected domain in the plane bounded by a simple closed curve, and proved the following results
Let be a simply connected bounded domain in the plane bounded by a simple closed curve. Assume has a straight portion . Let be an isoradial graph embedded into the whole plane with common radius , and be the largest subgraph of consisting of faces, and completely inside . Let be the interior dual graph of , and assume is the interior dual graph of . Assume has a straight part approximating . Let be the superposition of and with one boundary vertex of , approximating a point on as , removed. As the mesh size , the distribution of the height function for random perfect matchings on , in the scaling limit, is conformally invariant and converges to a Gaussian free field.
In particular, we do not assume the graph to be periodic, and we do not assume the boundary to be smooth except for a straight portion. Since the bipartite isoradial graph is obtained by the superposition of an arbitrary isoradial graph and its interior dual graph, the boundary conditions are discrete analogues of the Dirichlet boundary conditions and the Neumann boundary conditions. We prove, in this paper, the convergence of the inverse weighted adjacency matrix () for both boundary conditions. Although similar boundary conditions have been addressed on the uniform square grid ([13, 14]), the major difference of this paper from previous work lies in the fact that we are working on isoradial graphs, and the analysis on isoradial graphs are more complicated (). In particular, the discrete holomorphic functions on isoradial graphs are not Lipschitz in general. The is closely related to the local statistics of dimers (), and moreover, the spin-spin correlation of the Ising model ([10, 6]).
2.1 Isoradial graphs
In this subsection we review the definition of the isoradial graph as well as its basic properties. An isoradial graph is a graph which can be embedded into the plane such that each face is inscribable into a circle of common radius. The class of isoradial graphs includes the common regular graphs like the square grid, and hexagonal lattice, but is more general than that. See Figure 1 for examples of isoradial graphs.
An equivalent definition of the isoradial graph is the so-called rhombic tiling. Namely, we can always construct a planar graph from a planar graph , such that each face of is of degree 4. The vertices of are either vertices of , or faces of . Two vertices of are connected if and only if is a vertex of surrounding the face , or vice versa. Hence to see if a graph has an isoradial embedding, it suffices to see that if the constructed graph , in which each face is of degree 4, can be embedded onto the plane such that all edges have the same length, i.e., form a rhombic lattice.
In a planar graph with faces of degree 4, a train track is a path of faces (each face being adjacent along an edge to the previous face) which does not turn: on entering a face it exits across the opposite edge. It is proved in  that a planar graph with faces of degree 4 has a rhombic embedding if and only if no train track path crosses itself or is periodic; two distinct train tracks cross each other at most once.
In the above construction, each edge of corresponds to a rhombus in . An rhombus half angle is associated to each edge of the isoradial graph; it is the angle in formed by the edge and an edge of the corresponding rhombus of , see Figure 2. In Figure 2, the black edges are edges of the primal graph, the blue or red edges are edges of the dual graph, and dashed edges are edges of the rhombic lattice. From the picture it is quite clear that if the primal graph is isoradial, the dual graph is also isoradial.
A perfect matching, or a dimer configuration on an isoradial graph is a choice subset of edges such that each vertex is incident to exactly one edge.
2.2 Harmonic functions
The discrete Laplacian operator on isoradial graphs, maps a function defined on , to another function defined on as follows:
where is the rhombus half angle corresponding to the edge .
We can also associate a random walk, or a Markov chain to an isoradial graph, such that the transition probability
where is the probability that a random walk started at visits at the first step.
A discrete harmonic function on an isoradial graph is a function defined on vertices of the graph , satisfying
(1) can also be considered as a discrete analog of the mean value property.
The mean value property (1) obviously implies the maximal principle for discrete harmonic functions, i.e., if is a subgraph of the isoradial graph , and we define boundary vertices of to be vertices of that are incident to vertices outside , and interior vertices of to be vertices in that are incident only to vertices of . If is harmonic at any interior vertex of , then the maximal or minimal value of can only be achieved at boundary vertices of , if is not constant on all vertices.
The harmonic extension also has a discrete analog. If is a real-valued function defined on boundary vertices of , there exists a unique discrete harmonic function on such that the values of on boundary vertices of are the same as those of . In fact, can be written down explicitly as follows
for all , where is the probability that a random walk started at first hit the boundary at . is a harmonic function in and a probability measure on . If , is the exit probability through of the random walk started at .
2.3 Gaussian free field
The 2-dimensional Gaussian free field (GFF), or massless free field, is a natural 2-dimensional time analog of the Brownian motion.
Let be a simply-connected bounded domain of the complex plane . Let be the space of smooth, real-valued functions that are supported on compact subsets of . The GFF on , can also be considered as a Gaussian random vector on the infinite-dimensional Hilbert space , where is the completion of under the norm of derivatives, or more precisely, the GFF can be considered as a Gaussian random vector on the completion of with respect to a measurable norm ().
Let be an -orthornormal eigenfunctions for the Laplacian with Dirichlet boundary conditions (i.e., on ). Let be the eigenvalue of . The GFF on , is a random distribution (continuous linear functional) on functions of , such that, for any function ,
where are i.i.d. Gaussian random variables with mean 0 and variance 1.
3 Convergence of discrete holomorphic obzervables
In this section we introduce the discrete holomorphic observable for perfect matchings on isoradial graphs, namely, the so-called inverse Kasteleyan matrix entries, and prove its convergence in the scaling limit under special boundary conditions.
3.1 Discrete approximation
In this subsection we discuss the basic setting and assumptions for the discrete approximation, under which the convergence of discrete holomorphic observables will be proved.
Let be an isoradial graph on the whole plane, i.e., each face is inscribable in a circle of radius . Assume all the rhombus half-angles of are bounded uniformly away from and , i.e., assume there exists a constant , such that for all edges , the rhombus half angle . Let be a simply-connected bounded domain of the complex plane bounded by a simple closed curve. Let be an isoradial subgraph of , consisting of faces, i.e., each face of is inscribable in a circle of radius . Assume is also simply-connected, i.e., the boundary of , has only one connected component. Let be the interior dual graph of , and assume that is the interior dual graph of , as in Figure 2, where the primal graph is bounded by the outer boundary consisting of black edges, , the interior dual graph of , is the graph bounded by the outer boundary consisting of blue edges; and , which has as its interior dual graph, is the graph bounded by the outer boundary consisting of red edges.
Assume as , approximates in the following sense:
is the largest subgraph of , consisting of faces and satisfying .
Given , we construct another graph as follows. First of all, let be a graph with a vertex for each vertex, edge and face of , and an edge for each half-edge of and each half dual-edge as well. The graph is a bipartite planar graph, with black vertices of two types: vertices of and faces of . White vertices of corresponds to edges of , and dual edges in . According to the Euler’s theorem, has one more black vertices than white vertices. Let be obtained from by removing a black vertex on the boundary. Assume there exists , such that . Assume admits a perfect matching. (This is obviously true for the square lattice case, in which is the interior dual graph of a Temperleyn polyomino).
An isoradial embedding of gives rise to an isoradial embedding of . Each rhombus corresponding to an interior edge of with half-angle is divided into four congruent rhombi in . Each triangle (half-rhombus) corresponding to a boundary edge of is divided into two congruent triangles and one rhombus. The two triangles are part of a rhombus with half-angle , and the rhombus has half-angle . , as a subgraph of , inherited an isoradial embedding of , in which each face is inscribable in a circle of radius .
We define a symmetric matrix , indexed by vertices of , as follows: if and are not adjacent, . If and are adjacent vertices, being white and being black, then is the complex number of length , with direction pointing from to .
Let be obtained from by multiplying edge weights of around each white vertex (coming from an edge of of half-angle ) by , that is , where is the diagonal matrix defined by , and . Then an edge of coming from a “primal” edge of has weight for , and an edge coming from a dual edge has weight for . (There edges have weight , respectively for .) Let denote the transpose conjugate of .
Restricted to vertices of , we have , where is the Laplacian operator of with Neumann boundary conditions, and the boundary value at the removed vertex is . Restricted to faces of , we have , where is the Laplacian operator of , the interior dual graph of with Dirichlet boundary conditions.
Let be a black vertex of . Assume in , has neighbors with half-angle , then , where is the unit vector pointing from to . We have
If are adjacent vertices in , we have
and similar computations apply for the case when are adjacent vertices in .
If are at distance 2 in , but correspond to a vertex and a face of , we have two contributions to which cancel: in , let be the two edges from bounding the face , with on the right. Let be the two vertices in corresponding to . Then
Let be a function defined on the set of vertices of . First assume that is not adjacent to in . If is an interior vertex of , then the above computation gives
This is the same as the effect of discrete Laplacian operator on the whole-plane graph . If is a boundary vertex of , then the above computation gives
This is the same as the effect of discrete laplacian operator on the whole-plane graph , if we require that on all the vertices adjacent to in , has the same value as . If we consider the boundary of the domain consisting of dual edges of the adjacent edges of in , i.e., , this is the discrete analogue of normal derivative along the boundary.
Now assume is an adjacent vertex of in . After imposing the Neumann boundary conditions on vertices adjacent to , but not in , we have
This is the same as the effect of the discrete Laplacian on the whole-plane graph , by requiring that .
Now assume is a vertex of , and be a function defined on . We have
Similar argument shows that the effect of for on each vertex of will be the same as the discrete Laplacian of the whole plane graph for at the vertex of , if we require the value of on to be . ∎
The following lemma states the relation between the inverse matrix with the local statistics of the perfect matchings. It was first discovered for the hexagonal lattice in , then generalized to the isoradial graph in .
()The dimer partition function on is equal to . The probability of edges occurring in a configuration chosen with respect to Boltzmann measure is
3.2 Convergence with Dirichlet boundary conditions
From Lemma 2, we know that there are two types of entries, depending on whether the black vertex is a vertex of the primal graph or a vertex of the dual graph. In this subsection we deal with the convergence when the black vertex is a vertex of the dual graph. We start with a technical lemma proved in , which says that uniformly bounded sequence of harmonic functions, has a uniformly convergent subsequence on compact subsets of the domain.
()Let be (real-valued) discrete harmonic function defined on vertices of the isoradial graph with . If are uniformly bounded on , i.e.,
where is a constant independent of . Then there exists a subsequence , and two functions , , such that
if , and as . Moreover, the limit function , , is harmonic in and is analytic in .
The next lemma, also proved in , is a discrete analog of the Beurling-type estimate for the Brown motion in the continuous case. The proof is based on the approximation of the discrete Laplacian operator on isoradial graphs to the continuous Lapacian operator. It will be useful for us to identify the boundary value of harmonic functions.
Assume is a white vertex of , consider as a function of , where is a black vertex of . is adjacent to four vertices: two vertices corresponding to vertices of , denoted by and , and two vertices corresponding to faces of , denoted by and . Let denote the unit vector pointing from to , respectively. Let denote the rhombus half-angle corresponding to the edge . Explicit computations show that
Since the graph is periodic, is invariant under translations along the horizontal and vertical directions with finite periods, and there are finitely different orbits for edges under translations. Any two edges in the same orbit have the same weight and the same rhombus half-angle. We can actually classify all the white vertices of , which correspond to edges of by their orbits under the action of translations. We have the following lemma.
Let be a point in the interior of , and let , . Let be a white vertex of nearest to , which corresponds to an edge in with a fixed rhombus half-angle , and let be a vertex of nearest to , then
where is the Green’s function of the region , and the derivative is the directional derivative along the with respect to the first variable. Moreover, let be the diagonal set of , then the convergence is uniform for in compact subsets of .
Let be the corresponding matrix defined for the whole-plane graph , obtained by the superposition of and . Then
where is the discrete Green’s function on , the dual graph of on the whole plane. It is proved in  that
where is the Euler’s constant. Then
and the coefficient of in the error term is bounded uniformly for any in compact subsets of . Fix , and let
is a complex-valued function defined on vertices of , whose discrete laplacian on all vertices of vanishes. We consider and , the real and imaginary part of , separately. Both and are discrete harmonic on all vertices of , and the values of and on the boundary are bounded, and the bound is uniform, when is on the boundary, and is in compact subsets of . According to Lemma 5, and converge, in compact subsets of , to continuous harmonic functions and .
We also need to show that the boundary value of and are the limit of (3). Namely, we need to show that
Note that the right side of (4) and (5) are continuous in , and uniformly bounded when , and is in a compact subset of . Let be a ball centred at with radius , and let be the complement of the ball. Applying the harmonic extension formula (2), we have
According to the weak-Beurling type estimate in Lemma 5, we have
Let . Then converges to the function
This is times the directional derivative of the continuous Green’s function.
3.3 Near the straight boundary
Assume has a straight portion, denoted by . Assume has a straight portion approximating , denoted by . In this subsection we explore the behaviour of when one or both variables are near the straight boundary .
We can extend the isoradial graph across the flat boundary , such that the isoradial faces are symmetric with respect to the axis . We can also extend the Green’s function across as follows. If and are symmetric vertices with respect to , we define
Let be a point in , and such that . Using the extensions above, we have a discrete harmonic function in a neighborhood of , and they are uniformly bounded in the neighborhood of with a bound independent of . By Lemma 4, both the harmonic function and its directional derivatives converge uniformly in a neighborhood of .
3.4 Convergence with Neumann boundary conditions
In this subsection, we deal with the entries of where the black vertex is a vertex of the primal graph. It has a boundary condition which is a discrete analog of the Neumann boundary condition, given that the difference of observables along the normal direction of each boundary edge is 0. We will begin with a technical lemma in PDE which states that the derivatives of harmonic functions are bounded by themselves, and the bound could be uniform up to the boundary if the boundary has a smooth portion.
(Kellogg’s theorem and Schauder boundary estimate)Let be a bounded domain in with boundary, , , in , and on , where , then . Moreover,
See Theorem 6.19 in . ∎
In particular, Lemma 7 impiles that at the straight portion of , the continuous harmonic function have bounded second-order derivatives.
The following lemma states the convergence of when the black vertex is a vertex of the primal graph.
Let be a simply-connected, bounded domain bounded by a simple closed curve, and be an isoradial graph with common radius , such that is the largest subgraph of (the isoradial graph on the whole plane), consisting of faces of , lying completely inside . Let be a point in the interior of , and let and . Let be a white vertex of and be a vertex of . Assume as , approximates in the following sense
there exists , independent of , such that , and ;
as , the direction and rhombus half-angle of edges in corresponding to are fixed to be and .
Then fix , there exists a subsequence such that the real and imaginary parts of converge uniformly on compact subsets of to harmonic functions on , as . Let be the limit of , , and let be a sequence of black vertices approximating which have incident edges in with direction equal to a fixed vector , then
where (resp. ) is the directional derivative of the second variable with respect to the (resp. ) direction, for , .
Recall that there are two types of black vertices and , where corresponds to vertices of the graph , and corresponds to vertices of the dual graph .
Fix a white vertex . Define , as a function on vertices and , as a function on vertices as follows
Let be a vertex of . Let be two adjacent vertices of in , and be two adjacent vertices of in . Then
where is the operator on the whole-plane bipartite isoradial graph , namely, the superposition of and . Note that if , then . Let be unit vectors from to , we then have
where is the rhombus half-angle corresponding to the edge . Hence if , we have