Dimer packings with gaps and electrostatics
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Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved December 18, 2007 (received for review November 9, 2007)
Abstract
Fisher and Stephenson conjectured in 1963 that the correlation function (defined by dimer packings) of two unit holes on the square lattice is rotationally invariant in the limit of large separation between the holes. We consider the same problem on the hexagonal lattice, extend it to an arbitrary finite collection of holes, and present an explicit conjectural answer. In recent work we managed to prove this conjecture in two fairly general cases. The quantity giving the answer can be regarded as the exponential of the negative of the twodimensional electrostatic energy of a system of charges naturally associated with the holes. We further develop this analogy to electrostatics by presenting two different natural ways to define a field in our setup, and showing that both lead to the electric field, in the limit of large separations between the holes. For one of the fields, this is also stated as a limit shape theorem for random surfaces, with the continuum limit being a sum of helicoids. We conclude by explaining the relationship of our results to previous results in the physics literature on spin correlations in the Ising model.
Consider a lattice in the plane and regard it as a graph, with lattice points being vertices and lattice segments being edges. Any two vertices connected by an edge form a dimer. A dimer packing (also called dimer covering) of a lattice subgraph is a collection of dimers that cover each vertex exactly once.
There is a vast literature on dimer packings, both in mathematics and in physics. The dimer model considers a family {G_{n} }_{n≥1} of subgraphs of a lattice (which may carry weights on its edges) and some specific boundary conditions; it then focuses on determining the dimer packing partition function of G_{n} either exactly or asymptotically as n approaches infinity. This classical problem was solved for planar lattices by the Pfaffian method, independently by Kasteleyn (1) and by Temperley and Fisher (2, 3). A celebrated application is the solution of the twodimensional Ising model by rephrasing it as a dimer model (4, 5). Possibly the earliest entry in this literature is the enumeration of dimer packings of honeycomb graphs, obtained in an equivalent form by MacMahon (ref. 6; Sec. 429, Sec. 494) in the early twentieth century. A recent landmark paper is the work by Kenyon, Okounkov, and Sheffield (7), where planar bipartite lattices are classified according to the behavior of the variance of the height–height correlation function of the corresponding dimer models.
The interaction of gaps in dimer packings was introduced in the literature in 1963 by Fisher and Stephenson (8). Namely, let G_{n} be the subgraph of the square lattice ℤ^{2} induced by the vertex set {(i, j) : −n ≤ i, j < n}, and let m _{1} and m _{2} be two fixed vertices (“monomers”) having different colors in the chessboard coloring of the vertices of ℤ^{2}. The monomer–monomer correlation function “in a sea of dimers” is defined as where for a graph G we denote by M(G) the number of dimer coverings (equivalently, perfect matchings) of G.
Fisher and Stephenson conjecture that as one monomer stays fixed and the other recedes to infinity along any fixed direction, the monomer–monomer correlation function is asymptotically equal to Br ^{−1/2}, where r is the distance between the monomers and the constant B is independent of the direction (i.e., the monomer–monomer correlation is rotationally invariant in the limit of large separations). This conjecture still stands open today, four and a half decades after its formulation. The asymptotic behavior of this correlation function has been established only along one direction, that of a lattice diagonal (see ref. 9).
A More General Question, and a Conjectural Answer
In refs. 10–12, we managed to make progress on the Fisher–Stephenson rotational invariance conjecture by phrasing it on the hexagonal lattice and extending its scope to more (not just two) and larger (not just unit) holes. The general form remains open (see Conjecture 1), but two fairly general cases are proved: that of symmetric distributions of triangular holes of side two plus one additional monomer on the symmetry axis (see Theorem 2), and the case of arbitrary triangular holes of even side lengths (see Theorem 1).
Regard the hexagonal lattice ℋ from the point of view of its dual, the triangular lattice 𝒯. Draw 𝒯 in the plane so that one family of lattice lines is vertical. Then the vertices of ℋ are the unit triangles of 𝒯, and a dimer on ℋ is a unit rhombus (also called lozenge, or diamond) consisting of two unit triangles of 𝒯 that share an edge. Monomers on ℋ are unit triangles of 𝒯; we call them right monomers and left monomers according to the direction in which they point. Holes in ℋ are finite (not necessarily connected) unions of such monomers.
Call the midpoints of vertical lattice segments in 𝒯 marked points, and coordinatize them by pairs of integers in a 60° coordinate system by picking one of them to be the origin, and taking the x and y axes in the polar directions −π/3 and π/3, respectively. Then each right monomer is specified by a pair of integer coordinates, and so is each left monomer.
Define the right ktriangular hole ▷ _{k} (x, y) to be the rightpointing triangular hole with a side length of k units (or lattice spacings) whose topmost marked point (those on its boundary included) has coordinates (x, y); the left ktriangular hole ◁ _{k} (x, y) is defined to be the analogous leftpointing triangular hole. In some instances we will find it convenient to have a unifying notation for these two types of holes. To this end, for k ∈ ℤ we define the ktriangular hole Δ _{k} (x, y) by (see Fig. 1 for two illustrations).
Following in the spirit of ref. 8, we define the joint correlation function (for short, joint correlation, or simply correlation) ω̃ of any finite collection O _{1}, …, O_{n} of holes as follows. For any positive integer N, let T_{N} be the torus obtained from the rhombus {(x, y) : x, y ≤ N − 1/2} on 𝒯 by identifying opposite sides. Let the charge ^{†} q(O) of the hole O be the difference between the number of right and left monomers in O. By performing a reflection across a vertical lattice line, it suffices to define the correlation when Σ_{i=1} ^{n} q(O_{i} ) ≥ 0. Define ω̃ inductively by:

(i) If Σ_{i=1} ^{n} q(O_{i} ) = 0, set

(ii)If Σ_{i=1} ^{n} q(O_{i} ) = s > 0, set
where the constant C is determined by ω̃(▷_{1}(0,0),◁_{1}(R, 0)) ∼ C R ^{−1/2}, R → ∞.
Given a hole O and integers x and y, denote by O(x, y) the translation of O under which its topmost (and leftmost, if there are ties) marked point is brought to the point (x, y). Then the generalization of the monomer–monomer correlation problem posed by Fisher and Stephenson is the following.
Question.
Given a collection of hole types O _{1}, …, O_{n}, and the configuration obtained by placing them in positions O _{1}(x _{1}, y _{1}), …, O_{n} (x_{n} , y_{n} ), what is the asymptotics of as R → ∞?
Theorems 1 and 2 below lend support to the following answer.
Conjecture 1.
For any hole types O _{1}, …, O_{n} and any distinct pairs of integers (x _{1}, y _{1}), …, (x_{n} ,y_{n} ) we have as R → ∞ that where d is the Euclidean distance.
According to this, the asymptotics of the dimermediated interaction of holes on the hexagonal lattice captured by their joint correlation ω̃ is governed by the laws of twodimensional electrostatics. More precisely, it is given, up to a multiplicative constant, by the exponential of the negative of the electrostatic energy of the twodimensional system of physical charges obtained by viewing each hole as a point charge of magnitude and sign specified by the statistic q. The close analogy with electrostatics is discussed in detail in refs. 11 and 12.
For other lattice models in the physics literature that are equivalent to a Coulomb gas, see the comprehensive survey of Nienhuis (13).
Remark 1:
From this point of view, the rotational invariance of the monomer–monomer correlation conjectured by Fisher and Stephenson emerges as a special case of the analog of this electrostatic phenomenon on the square lattice. Indeed, for the square lattice analog of Conjecture 1, q(O) is defined to be the difference between the number of white and black unit squares in a chessboard coloring that fall inside the hole O (evidence that the square lattice analog of Conjecture 1 holds is presented in ref. 11). Since the two monomers have charges equal to 1 and −1, respectively, the formula above becomes precisely Br ^{−1/2}, where r is the Euclidean distance between the monomers, and B is some constant independent of the slope of the straight line connecting them.
Evidence for Conjecture 1
Under the additional assumption that the total charge of the holes is even, define a variant ω̄ of the above correlation ω̃ inductively by (i) and the modification of (ii) in which ◁_{1}(R, 0) is replaced by ◁_{2}(R, 0) (note that this causes the constant C to be replaced by the leading coefficient C′ in the asymptotics of ω̄(▷_{2}(0, 0), ◁_{2}(R, 0)), R → ∞; it turns out that C′ = 3/4π^{2}, and ω̄(▷_{2}(0, 0), ◁_{2}(R, 0)) ∼ 3/4π^{2} R ^{−2}).
The following results which we proved in ref. 12 (Theorem 2.1) and ref. 11 (Theorem 2.1) are evidence toward the statement of Conjecture 1. In independent work, Krauth and Moessner (14) predicted, based on Monte Carlo simulations, that the square lattice analog of Conjecture 1 holds for the case of two monomers (of not necessarily opposite colors).
A hole is called pure if it is the disjoint union of 2triangular holes of the same type: either all ▷_{2}'s, or all ◁_{2}'s (the constituent ▷_{2}'s or ◁_{2}'s are not required to be contiguous). A pure hole is linear if the centers of its constituent ▷_{2}'s or ◁_{2}'s are collinear (see Fig. 2 for two examples). We say that the slope q of a pure and linear hole is admissible if the numerator of the lowest terms representation of 1 − q is a multiple of 3. The following is a restatement of Theorem 2.1 of ref. 12.
Theorem 1.
For any pure and linear holes O _{1}, …, O_{n} of admissible slopes, and any distinct pairs of integers (x _{1}, y _{1}), …, (x_{n} , y_{n} ), we have
Our proof of this theorem (see ref. 12) is based on a determinant formula due to Kenyon (15, 16), which extends expressions found by Fisher and Stephenson (see ref. 8) for the dimer–dimer correlation on the square lattice, and is in turn based on the Pfaffian method. When Σ_{i=1} ^{n} q(O_{i} ) = 0, this formula expresses the correlation ω̄(O _{1}, …, O_{n} ) as a certain determinant whose entries are Fourier coefficients of a simple twovariable function (this is called the coupling function, or Green's function). We extend the determinant formula to the case of arbitrary total charge and, by a sequence of combinatorial and analytic arguments, deduce that its asymptotics is given by the expression above.
Recall that Δ _{k} stands for ▷ _{k} if k is a positive integer, and for ◁_{−k} if k is a negative integer.
Corollary (12).
For any even integers k _{1}, …, k_{n} and any distinct pairs of integers (x _{1}, y _{1}), …, (x_{n} , y_{n} ) we have
In ref. 11 we consider another definition for the correlation of holes on the triangular lattice. Namely, given holes O _{1}, …, O_{n} , set k = Σ_{i=1} ^{n} q(O_{i} ) and denote by H_{N,k} the lattice hexagon on 𝒯 centered at the origin and having side lengths alternating between N + k and N (with the side on the left having length N + k). Let the collection O _{1} ^{(0)}, …, O_{n} ^{(0)} consist of translations of the given holes into some fixed “reference” positions. Then the correlation ω of ref. 11 is defined by In ref. 11 we prove the following result.
Theorem 2.
Let T_{1}(x _{1}, y _{1}), …, T _{n} (x_{n} , y_{n} ) be a collection of triangular holes on 𝒯 that is symmetric with respect to a horizontal symmetry axis ℓ, contains precisely one hole of type ▷_{1} or ◁_{1} (necessarily placed symmetrically across ℓ), and has all remaining holes of type ▷_{2} or ◁_{2}. Assume also that all T_{i}'s to the left of the unit hole are of type ◁_{2}, and all T_{i}'s to the right of it are ▷_{2}'s. Then where c_{m,n} is some explicit constant depending only on m, n, and the choice of the reference position of the holes.
Our proof of Theorem 2 (see ref. 11) is based on a certain multiparameter extension we found in ref. 17 of MacMahon's classical enumeration of lozenge tilings of lattice hexagons on 𝒯. In the special case when all the holes line up along the symmetry axis, our product formulas [ref. 17, equations (1.1)–(1.9)] lead directly (after straightforward if lengthy manipulations) to the statement above. In the general symmetric case, the correlation is expressed as a multiple sum of terms, each corresponding to an instance of collinear holes. This multiple sum is then analyzed by using a variety of combinatorial and analytic arguments, and shown to have the stated asymptotics. This proof does not involve the Pfaffian method.
The FField
In the present article we further the analogy with electrostatics by presenting two natural ways to associate a field to a configuration of holes, both of which turn out to lead to the twodimensional electric field of the corresponding physical system of charges. The details are given in ref. 20 for the first field.
To define the first field, fix a placement O _{1}, …, O_{n} of holes on the triangular lattice, and a left monomer e outside them. Pick N large enough so that both the holes and e are enclosed in the rhombus {(x,y): x, y ≤ N − 1/2}, and denote as before by T_{N} the torus obtained from this rhombus by identifying opposite sides. Assume Σ_{i=1} ^{n} q(O_{i} ) = 0, and let X _{N} be the random variable returning for each lozenge tiling of T_{N} \ O _{1}∪…∪O_{n} the unit vector pointing from the center of e along the long diagonal of the lozenge that covers e. Set (the notation marks the fact that this definition was inspired by Feynman sums). This defines the discrete field F at the center of each leftmonomer. The definition is easily extended to the case of arbitrary total charge, by the inductive approach employed in the definition of the correlations ω̃ and ω̄ (see ref. 20). The field obtained in this way for two oppositely oriented holes of side two is pictured in Fig. 3(some long vectors close to the holes have been omitted for clarity). A sidebyside comparison reveals it to be very close to the twodimensional Coulomb field produced by two charges of magnitudes 2 and −2.
Theorem 3.
Let k _{1}, …, k_{n} ∈ ℤ be even, and (x _{0}, y _{0}), …, (x_{n} , y_{n} ) ∈ ℝ^{2} be distinct. Then for any sequences of integers {x_{i} ^{(R)}} _{R} and {y_{i} ^{(R)}} _{R} with we have
Remark 2:
Regard the arguments on the lefthand side above as residing on the lattice 𝒯 _{R} obtained from 𝒯 by a homothety of modulus 1/R based at the origin [a homothety, also called similarity, based at the origin, is a map of the form (x, y) ↛ (αx, αy) of the plane onto itself; α is its modulus]. Then by the assumptions of the theorem, the geometrical image of the point (Rx _{0} ^{(R)}, Ry _{0} ^{(R)}) converges to (x _{0}, y _{0}), and for each i = 1, …, n hole Δk_{i} (Rx_{i} ^{(R)}, Ry_{i} ^{(R)}) shrinks down to point (x_{i} , y_{i} ), as R → ∞.
𝒯 _{R} is a unit triangular lattice in the coordinate system _{R} 𝒞 obtained from our 60° system by shrinking units to 1/R of their lengths. If a point has coordinates (x, y) in our original coordinate system, denote by ( _{R}x, _{R}y) its coordinates in _{R} 𝒞; we clearly have ( _{R}x, _{R}y) = (Rx, Ry).
Since q(Δ _{k} ) = q(▷ _{k} ) = k for positive k and q(Δ _{k} ) = q(◁_{k}) = − k = k also for negative k, the sum on the righthand side in Eq. 13 equals As _{R}𝒞 is a 60° coordinate system, the Euclidean distance between two points is given by Then the sum in Eq. 14 becomes Thus, Eq. 13 viewed on 𝒯 _{R} states that in the limit as R → ∞, the field F created by a set of triangular holes is given, up to a constant multiple, by the twodimensional electrostatic field of the system obtained by regarding each hole Δ _{k} as a point charge of magnitude q(Δ _{k} ).
Proof outline:
It follows from the definition that for any left monomer e, where L _{1}, L _{2}, and L _{3} are the three lozenges that cover e, while e _{1}, e _{2}, and e _{3} are the unit vectors pointing from the center of e along their long diagonals. We need to show that when the coordinates of the holes are as in the statement of the theorem, the asymptotics of the righthand side of Eq. 17 is given by the expression on the righthand side of Eq. 13 .
Suppose the set of monomers in a given collection 𝒪 of holes can be partitioned into pairs sharing at least one common vertex (this is the case for all hole collections in Eq. 17 ). If the total charge of the collection of holes 𝒪 is 0, Kenyon's formula (15, 16) expresses the correlation ω̄(𝒪) as where (r, r′) ranges over the right monomers in the union of holes, and (l, l′) ranges over the left monomers in the union of holes. An extension of this that we found in ref. 12 applies also in the case of nonzero total charge.
Thus, by Eq. 17 the field F can be written in terms of determinants of “P matrices” of the type above. When the coordinates of the holes are as in Eq. 13, the entries of these matrices can be written as Laplace integrals, which allows finding their asymptotics. One obstacle that needs to be surmounted is that the matrices consisting of the leading terms of these P matrices are singular in general. We resolve this by performing convenient row and column operations, which result in matrices whose leading terms constitute matrices of nonzero determinants. Then Eq. 13 reduces to proving that certain combinatorial determinants evaluate to the the components of the righthand side of Eq. 13 . An interesting feature of these determinant evaluations is that, unlike the vast majority of the examples in the literature (see, e.g., refs. 18 and 19) that have their value given by a product, they involve sums of products. We manage to deduce them from a determinant evaluation proved in ref. 12.
Random Surfaces
Each lozenge tiling of a simply connected region on the triangular lattice can be viewed as the the upper surface of a stack of unit cubes: simply use the “visual lifting” that regards one of the two lozenge tilings of a hexagon of side 1 as a unit cube that is in the stack, and the other as a missing unit cube (Fig. 4).
An important problem in this context is to study the scaling limit behavior of the surfaces (also called height functions) resulting this way from tilings of a region. More precisely, fix a simply connected region ℛ in the plane, and analyze the height functions corresponding to tilings of ℛ ∩𝒯_{ϵ} in the limit ϵ → 0, where 𝒯_{ϵ} is the image of the triangular lattice 𝒯 under a homothety of modulus ϵ based at the origin. Landmark examples in the literature include the determination of the typical shape of a boxed plane partition by Cohn, Larsen, and Propp (21) and the recent classification of the dimer models on planar bipartite lattices according to the behavior of the variance of the height–height correlation function by Kenyon, Okounkov, and Sheffield (7).
This lifting to a surface breaks down for lattice regions with holes of nonzero charge (see Fig. 5). However, even in the presence of such holes one can lift each tiling to a multisheeted surface; this is illustrated in Figs. 5 and 6 (see ref. 20 for details). Averaging over such surfaces can also be defined. Thus, the following question arises: given a set of triangular holes on 𝒯, what is the behavior of the average of the the multisheeted surfaces arising from tilings of their complement, as the lattice spacing approaches zero?
The answer turns out to be given in terms of halfhelicoids (see ref. 22 for an overview of a variety of other settings in which helicoids arise). The limit of the average of lifting surfaces as on the right of Fig. 6 is a surface of the type illustrated in Fig. 7.
The halfhelicoid H ^{+}(a, b; c) is the surface whose parametric equations in Cartesian coordinates are Given a positive integer s, define the srefined halfhelicoid by (+α on the righthand side indicates translation by α along the z axis).
Note that for s ∈ ℤ, the fibers of H _{s} ^{+}(a, b; sc) above each point u in the xy coordinate plane are of the form f(u) + 2πcℤ. Thus, given H_{si} ^{+}(a_{i} , b_{i} ; s_{i}c), s_{i} ∈ ℤ, i = 1, …, k, one can define their sum by defining the fiber of S above u to be In ref. 20 we prove the following result.
Theorem 4.
Let k _{1}, …, k_{n} ∈ ℤ be even, and (x _{0}, y _{0}), …, (x_{n} , y_{n} ) ∈ℝ^{2} be distinct. Let {x_{i} ^{(R)}} _{R} and {y_{i} ^{(R)}} _{R} be sequences of integers so that Consider the holes Δk_{i} (x_{i} ^{(R)}, y_{i} ^{(R)}), i = 1, …, n, on the triangular lattice 𝒯 _{R} with lattice spacing 1/R, lift the tilings of their complements to multisheeted surfaces, and take their average: Then, as R → ∞, RS _{av} ^{𝒯R} converges to the sum of refined halfhelicoids For any bounded set B and any open set U containing (x _{1}, y _{1}), …, (x_{n} , y_{n} ), the convergence is uniform on B \ U.
The TField
The next construction is inspired by the standard way of defining the electric field in physics, by the force exerted on a test charge. Let O _{1}, …, O_{n} be a fixed placement of holes on the triangular lattice. For any x, y, α, β ∈ ℤ define [the hole ▷_{2}(x, y) plays the role of a “test charge,” and the T _{α,β}(x, y)'s record the effects of its displacements; technical reasons in the proof of Theorem 5 require the test hole to be a ▷_{2} rather than a ▷_{1}]. The form of the expression in parentheses was suggested by the fact that it is the discrete directional derivative of lnω̄, and as pointed out in the comments following Conjecture 1, lnω̄ is the analog of the electrostatic potential energy. We can prove the following result.
Theorem 5.
(a) Let k _{1}, …, k_{n} ∈ ℤ be even, (x _{0}, y _{0}), …, (x_{n} , y_{n} ) ∈ ℝ^{2} distinct, and α, β ∈ ℝ. There exists a unique vector field T on ℝ^{2} \ {(x _{1}, y _{1}), …, (x_{n} , y_{n} )} so that for any integer sequences {x_{i} ^{(R)}} _{R} , {y_{i} ^{(R)}} _{R} , {α^{(R)}} _{R} , and {β^{(R)}} _{R} with lim_{R→∞} α^{(R)}/R = α, lim_{R→∞}β^{(R)}/R=β, and we have where proj _{v}u denotes the component of the vector u in the direction of the vector v. (b)
Remark 3:
We can also prove that if we replace the test hole ▷_{2} in the definition of the T _{α,β} (x, y)'s by any pure linear hole, the statement of Theorem 5 holds without change.
Democratic Electrostatics
The definition of both the Ffield and the Tfield given above can readily be extended to any planar lattice. Despite their asymptotic equality (up to a constant multiple) on the hexagonal lattice (expressed by Theorems 3 and 5), it turns out that in general these two fields can have very different asymptotic behaviors. Next we present an example that illustrates this, which we found recently in joint work with D. B. Wilson (Microsoft Research, Redmond, WA).
The critical Fisher lattice ℱ is the planar lattice of equilateral triangles and regular dodecagons in which the edges of the triangles have weight 1, and the intertriangle edges have weight
Given any set of antiholes on ℱ, we define their correlation ω_{ℱ} by enclosing them in large rectangular lattice regions, turning them into tori, considering the ratio of the dimer coverings of the tori with the antiholes present versus them being absent, and taking the limit as the tori grow to infinity.
Based on extensive exact calculations of specific antihole correlations, assumptions of conformal covariance, and numerical analysis arguments, we were led to guess the following formula. Let ℱ _{R} be the lattice obtained from ℱ by a homothety of modulus 1/R about the origin, and consider sequences {a _{1} ^{(R)}} _{R} , …, {a _{2k} (R)} _{R} , where the a _{i} ^{(R)}'s are antiholes on ℱ _{R} , and Then we have checked numerically to very high precision that the formula holds with C ≈ 0.9587407138742449.
Interestingly (and unbeknown to us while we were doing the calculations that led to Eq. 31 ), exactly the same formula appeared before in the physics literature, in the context of the Ising model. Using the analogy of the Ising problem to a twodimensional fermion field theory, Luther and Peschel (23) deduced that the scaling limit of the spin correlation of 2k spins in the Ising model on the square lattice at critical temperature is given by the righthand side of Eq. 31 [the first occurrence of the explicit general form of this seems to be in the work of Richardson and Bander (24); the case of collinear spins was addressed by Zuber and Itzykson (25)].
This coincidence can be explained in outline as follows. By using the hightemperature expansion, one can show that in the Ising model on the vertices of the hexagonal lattice, the spin correlation of the 2k spins at vertices v _{1}, …, v _{2k} can be expressed as the partition function of loopstring configurations on the hexagonal lattice, with the set of endpoints of strings being precisely {v _{1}, …, v _{2k}} (we learned about this equivalence from ref. 26). In turn, such loopstring configurations can be shown to be in onetoone correspondence with dimer coverings of the Fisher lattice ℱ with an antihole in each of the triangular “cities” corresponding to v _{1}, …, v _{2k} (an analogous construction for the square lattice is given in ref. 26). Note also that the latter can be identified with the union of 3^{2k} families, each consisting of the dimer coverings of ℱ with 2k unit holes, one in each of the triangular cities corresponding to v _{1}, …, v _{2k} (this follows since in a dimer covering the central vertex of each antihole must be matched to one of its neighbors).
Remark 4:
If we partition the set of antiholes into two equally sized parts, view the antiholes in the first set as positive unit charges, the others as negative unit charges, write down the exponential of the negative of their twodimensional electrostatic energy, and sum over all such partitions, we get precisely the quantity under the square root on the righthand side above. Based on this, we refer to the interaction described by Eq. 31 as democratic electrostatics.
Remark 5:
As far as the Ffield created by a set of antiholes is concerned, data consisting of exact calculations strongly suggest the surprising fact that it scales to a field whose vectors are all parallel to one another. On the other hand, the formula in Eq. 31 for the scaling limit of ω_{ℱ} readily yields a conjecture for the Tfield created by a set of antiholes. The latter is very different from a field of parallel vectors.
The equivalence described above between the two viewpoints that led to Eq. 31 elucidates also the relationship between our results stated in Theorems 1 and 2 and the results in the substantial physics literature on the asymptotics of spin correlations in the Ising model [some landmarks of which are the pioneering work of Onsager (27), the elegant alternative derivation of Montroll, Potts, and Ward (28), the classic text of McCoy and Wu (29) and the abovementioned formulas for multispin correlations]. Namely, while the latter implies that the dimermediated interaction of unit holes on the critical Fisher lattice is governed by democratic electrostatics, the former proves that the analogous interaction of holes on the hexagonal lattice is governed by the usual twodimensional electrostatics.
Acknowledgments
This work was supported in part by National Science Foundation Grant DMS 0500616.
Footnotes
 *To whom correspondence should be addressed. Email: mciucu{at}indiana.edu

The author declares no conflict of interest.

This article is a PNAS Direct Submission.

↵ † To be in accord with physics usage, we use in this article the notation q(O) for the charge of a hole O. Note that this was denoted by ch(O) in refs. 11, 12, and 20.

Freely available online through the PNAS open access option.
 © 2008 by The National Academy of Sciences of the USA
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