INTERACTIONS BETWEEN INTERLEAVING HOLES IN A SEA OF UNIT RHOMBI TOMACK GILMORE 6 Abstract. Consider a family of collinear, equilateral triangular holes of any even 1 side length lying within a sea of unit rhombi. The results presented below show 0 that as the distance between the holes grows large, the interaction between them 2 may be approximated, up to a multiplicative constant, by taking the exponential b of the negative of the electrostatic energy of the system obtained by viewing the e holes as a set of point charges, each with a signed magnitude given by a certain F statistic. Furthermoreitisshownthattheinteractionbetweenafamilyofleftpointing 2 collineartriangularholesandafreeboundarymaybeapproximated(againuptosome multiplicative constant) by taking the exponential of the negative of the electrostatic ] h energy of the system obtained by considering the holes as a set of point charges and p theboundaryastraightequipotentialconductor. Thesetwodifferingsystemsofpoint - h charges can be related via the method of image charges, a well-known physical law t thatalsosurfacesinthefollowingmathematicalanalysisofenumerationformulasthat a m count tilings of certain regions of the plane by unit rhombi. [ 2 v 1. Introduction 5 6 Interactions between holes (or gaps) in two dimensional dimer systems were first 9 considered by Fisher and Stephenson, whose seminal paper [11] examined three types of 1 0 interaction: theinteractionbetweentwodimers; theinteractionbetweentwomonomers; . 1 and the interaction between a dimer and a fixed boundary (that is, an edge or a corner). 0 While this work focused exclusively on interactions between holes in the square lattice, 6 1 Kenyon [15] later generalised the first of these interaction types to an arbitrary number : of dimer gaps on both the square and hexagonal lattices. Kenyon, Okounkov, and v i Sheffield[16]thenextendedtheseresultsevenfurthertoincludegeneralbipartiteplanar X lattices. r a Interactions between non-dimer gaps on the hexagonal lattice (in particular gaps consisting of a pair of monomers) have been studied extensively by Ciucu [3][4][5][6], establishing therewith close (conjectural) analogies between such interactions and two dimensional electrostatic phenomena. More specifically, Ciucu conjectures that the asymptotic interaction between non-dimer holes in a two dimensional dimer system on the hexagonal lattice is, up to a multiplicative constant, inversely proportional to the product of the pairwise distances between the holes raised to some exponential power that is determined by each pair of holes. Such an interaction is said to be governed by Coulomb’s law for electrostatics since it may be obtained by taking the exponential of the negative of the electrostatic energy of the two-dimensional system of physical charges obtained by considering the holes as point charges of a certain magnitude and sign. A more formal statement of this conjecture together with a discussion of the evidence in support of it may be found in Ciucu’s excellent survey paper [7]. Although it has been shown to hold for a very general class of holes in dimer systems that are 1 2 T. GILMORE Figure 1. Part of a rhombus tiling of the plane, containing a set of horizontally collinear triangular holes of even side lengths where the sum of the charges of each hole is zero. embedded on tori [5] and also for a certain class of holes in planar dimer systems [4], a complete proof of the conjecture remains elusive. Further to the above mentioned interactions, a wholly (or perhaps, hole-ly) new type of non-dimer interaction was presented more recently in [9]: that of the interaction between a triangular hole and a so-called “free” boundary. Such an interaction also parallels certain physical phenomena, namely it appears to behave in analogy to the attraction of an electric charge to a straight line conductor. The author of the present work showed in [13] that similar behaviour may also be observed for a triangular hole that has been rotated 180◦ and thus points toward the boundary. Somewhat mysteri- ously, it seems the orientation of the hole has a direct effect on the interaction between the hole and the free boundary (a physical interpretation of this discrepancy has yet to be realised). Thecontributionofthecurrentarticleistwofold. Firstly,theinteractionofanentirely new class of triangular holes is established, namely the interaction between interleaving holes of any even side length that lie along a horizontal line within a sea of unit rhombi (here interleaving means that all holes that point in one direction do not necessarily all lie to one side of all holes that point in the opposite direction, see Figure 1). It would appear that aside from the aforementioned results obtained by embedding tilings on tori [5], such interactions for the planar case have yet to be treated in the literature. Theorem 1 below shows that under certain conditions the planar case does indeed agree with Ciucu’s tori result, however the results stated in Section 4 show that this type of interaction depends somewhat delicately on the rate at which the boundaries of the plane approach infinity. Secondly, the interaction between a set of left pointing triangular holes and a free boundary is established, thus generalising the earlier work of Ciucu and Kratten- thaler [9] to include any (finite) number of left pointing triangular holes of any even side length. Once again it is shown that such interactions are in some sense governed by Coulomb’s law, since they may be approximated by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing the holes as a set INTERACTIONS BETWEEN INTERLEAVING HOLES IN A SEA OF UNIT RHOMBI 3 Figure 2. Adimercoveringofaholeysubregionofthehexagonallattice, left, and the corresponding rhombus tiling on the triangular lattice, right. The unit triangles in red on the right correspond to the vertices that have been removed from the interior of the subgraph on the left. of point charges and the free boundary a straight equipotential conductor. According to the method of images [10, Chapter 6] the electrostatic energy of such a system is half the electrostatic energy of the system obtained by replacing the conductor with imaginary charges (these charges of opposite signed magnitude are obtained by reflect- ing the original charges through the conductor). One sees this well-known physical law surfacing through the purely mathematical analysis of certain enumeration formulas that count tilings of certain regions of the plane by unit rhombi, thus adding further support to the on-going electrostatic program of Ciucu. 2. Set-up and Results The main results presented in this article concern interactions between holes in two dimensional dimer systems. In the spirit of Fisher and Stephenson [11], suppose R is n a subgraph (with size parametrised by n) of some two dimensional lattice with a fixed set of vertices (indexed by the set H) removed from its interior. Denote this region containing a set of holes R \H. A perfect matching between vertices in R \H is also n n known as a dimer covering of R \H, and the set of all coverings of this region is known n as a dimer system. As n tends to infinity, dimer coverings of R \ H become dimer n coverings of the entire plane (in other words, sending n to infinity yields a set of holes that lie within a sea of dimers, see Figure 1), and the interaction between the fixed holes (otherwise known as the correlation function of the holes) in the dimer system is defined to be M(R \H) n ω (H) = lim , (2.1) R n→∞ M(Rn) where M(R) denotes the number of dimer coverings (equivalently, perfect matchings) of the region R. This paper focuses on dimer systems on the planar hexagonal lattice, 4 T. GILMORE O Figure 3. The holey hexagon H{−2,6},{0,−8}, left, and a rhombus tiling 10,4 of the same region, right. H , considered in terms of its “dual”, that is, the planar triangular lattice consisting of unit triangles, T , drawn so that one of the families of lattice lines is vertical. In this context a matching between two neighbouring vertices of H corresponds to joining a pair of unit triangles in T that share precisely one edge, therefore a dimer covering of H (from which a finite number of vertices may have been removed) corresponds to a tiling of the plane by unit rhombi (where a corresponding set of unit triangles have been removed). An example of a dimer covering of a subregion of H and its corresponding rhombus tiling on T may be found in Figure 2. It was conjectured by Ciucu [7] in 2008 that the interaction between any set of holes, H, that lie far apart within a sea of unit rhombi is asymptotically equal to (cid:89) (cid:89) ω˜(h) d(hi,hj)12q(hi)q(hj), h∈H 1≤i<j≤|H| where ω˜(h) is a constant dependent on each individual hole, q(h) is the charge1 of the hole h, and d(h ,h ) is the Euclidean distance between the holes h and h indexed by i j i j H. Although this conjecture remains open, the main result of this paper shows that it holds for a general family of holes that appear to have not yet been considered in any of the literature, namely triangular holes of any even side length that are horizontally collinear (that is, they lie on a horizontal line about which they are symmetrically distributed), where the sum of the charges of the holes is zero.2 An example of such a family of holes may be found in Figure 1. Theorem 1. The interaction between a set H of horizontally collinear triangular holes of any even side length is asymptotically (cid:89) (cid:89) ωH(H) ∼ Ch d(hi,hj)21q(hi)q(hj) h∈H 1≤j<i≤|H| 1The charge of h is a statistic on the hole given by the number of right pointing unit triangles that comprise it minus the number of left pointing ones. For example a right pointing triangular hole of side length two has charge 2, whereas a left pointing hole of the same size has charge −2. 2It shall be assumed that all sets of holes considered in this paper satisfy this condition. INTERACTIONS BETWEEN INTERLEAVING HOLES IN A SEA OF UNIT RHOMBI 5 Figure 4. A set of contiguous triangular holes of side length two where the forced unit rhombi are coloured grey, left, and the larger induced hole of side length six, right. as the distance between the holes h and h in H becomes large, where i j 1|q(h)|−1 2 (cid:89) 3s+1/2 C = Γ(s+1)2. h 2π s=0 Remark 1. The above theorem shows that the interaction between sets of interleaving, collinear triangular holes of any even side length may be approximated, up to a multi- plicative constant, by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing each hole as a point charge with signed magnitude given by the statistic q. A more detailed discussion of the close analogies between rhombus tilings of regions containing holes and certain electrostatic phenomena may be found in [7] and [9]. In order to prove Theorem 1 an exact formula (Theorem 3) is established in Section 3 that counts rhombus tilings of a holey hexagon3 centred at some origin, O, with sides of length n,2m,n,n,2m,n (going clockwise from the southwest side), containing p-many left and (p-many) right pointing horizontally collinear triangular holes of side length two lying along the horizontal line that intersects the origin. Such a region is denoted HL,R , where R = {r ,...,r } and L = {l ,...,l } are sets of unique integers that n,2m 1 p 1 p correspond to lattice distances of the midpoint of the vertical sides of the right and left pointing holes (respectively) from O. An example of a holey hexagon may be found in Figure 3. Remark 2. A string of k-many contiguous4 triangular holes of side length two is equiv- alent to a triangular hole of side length 2k, since dimers are forced within the “folds” of the holes (see Figure 4), thus inducing a larger hole. It is therefore sufficient to consider holey hexagons containing holes of side length two, and holes of a larger even side length shall be referred to as induced holes. 3This term was first coined by Propp in [20] to describe a hexagonal region that contains a set of holes in its interior. 4A set of horizontally collinear holes are contiguous if no horizontal rhombi can fit between them. 6 T. GILMORE Figure 5. TheregionsH(cid:98){−2,6},{0,−8}, upperleft, andHq{−2,6},{0,−8}, lower 10,4 10,4 left, together with tilings of each region, right, where the pairs of yellow tiles have a combined weight of 2. Theorem 3 follows from splitting HL,R into two subregions, each obtained by cutting n,2m alongthezig-zaglinethatproceedsjustbelowthehorizontallineintersectingtheorigin. The upper region is denoted H(cid:98)L,R , the lower HqL,R . According to Ciucu’s factorisation n,2m n,2m theorem [2], M(HL,R ) = M(HqL,R )·M (H(cid:98)L,R ), (2.2) n,2m n,2m w n,2m where M (H(cid:98)L,R ) denotes the weighted count of tilings of H(cid:98)L,R , where every pair of w n,2m n,2m unit rhombi that lie within the “folds” of the lower zig-zag boundary have a combined weight of 2 (see Figure 5). Exactenumerativeformulas(Theorem5andTheorem7)thatcount(weighted)tilings of these subregions follow from translating tilings to families of non-intersecting lattice paths in the usual way. According to [12], enumerating such families of paths amounts to evaluating two determinants and the product of these determinant evaluations then yields Theorem 3 by way of Ciucu’s factorisation result (2.2). Enumerating tilings of HL,R in this way also gives, for free, two enumeration for- n,2m mulas for certain symmetry classes of tilings of HL,R . It should be clear that tilings n,2m of HqL,R correspond to horizontally symmetric tilings of HL,R . Moreover, if each r in n,2m n,2m R is positive and satisfies r = −l for some l in L, then according to Ciucu and Krat- tenthaler [8] the weighted count of tilings of the upper region, M (H(cid:98)L,R ), is equal to w n,2m the number of vertically symmetric tilings of HL,R , which correspond to tilings of the n,2m left half of HL,R constrained on the right by a vertical free boundary that intersects n,2m the origin (here a boundary is considered free if unit rhombi are permitted to protrude halfway across it). Such a region shall be denoted VL and an example of a tiling of n,2m such a region may be found in Figure 6. Tilings of VL correspond to tilings of the left half of the plane constrained on the n,2m rightbyaverticalfreeboundaryasnandmaresenttoinfinity. Thecorrelationfunction of holes that lie within tilings of this half plane may be interpreted as the interaction between a set of left pointing holes and a vertical free boundary (see Figure 6, right). INTERACTIONS BETWEEN INTERLEAVING HOLES IN A SEA OF UNIT RHOMBI 7 Figure 6. AverticallysymmetrictilingoftheholeyhexagonHL,R where 12,4 L = {−8,−4,−2}andR = {2,4,8},left,togetherwiththecorresponding tiling of VL , centre, and a set of left pointing holes in a sea of dimers 12,4 constrained on the right by a free boundary, right. Ciucu and Krattenthaler [9] considered such an interaction for a single left pointing hole of side length two, which is a special case of the following theorem. Theorem 2. The interaction between a set of horizontally collinear left pointing holes of any even side length in a sea of unit rhombi and a right vertical free boundary is (cid:89) (cid:89) ωV(H) ∼ Kh d(hi,hj)14q(hi)q(hj), h∈H 1≤j<i≤|H| where H indexes both the left pointing holes and their reflections in the free boundary, and 1|q(h)|−1 2 (cid:89) 3s/2Γ(s+1) K = √ . h 2π s=0 Remark 3. The above result is in fact the square root of the result given in Theorem 1, and indeed this relationship is analogous to well-known and established physical laws. Considerapointchargepwithsignedmagnitudesituatednearastraightlineequipoten- tial conductor. The method of images [10, Chapter 6] states that in order to calculate the electrostatic energy of the system one may replace the straight line conductor with an imaginary charge, pˆ, of opposite signed magnitude to that of p situated at the po- sition specified by reflecting p through the conductor since the electric field induced by the charge(s) is the same for both arrangements (see Figure 7 for an illustration of this principle). The electrostatic energy of the system with the conductor is then one half of the electrostatic energy of the system consisting of p and its imaginary counter- charge pˆ. Thus the result above may be obtained, up to a multiplicative constant, by taking the exponential of the negative of half the electrostatic energy of the the system obtained by considering the set of left pointing holes described above together with their reflections through the vertical free boundary as a set of point charges with 8 T. GILMORE Figure 7. A diagram displaying the method of images, taken from [22]. The electric field induced by the positive charge, p, near the horizontal straight line conductor is the same as the electric field induced by the p anditsimaginarychargepˆobtainedbyreflectingpthroughtheconductor. signed magnitude given by q. One should see that this is equivalent (again up to some multiplicative constant) to taking the square root of the result given in Theorem 1. It should be noted that Theorem 1 and Theorem 2 are specialistions of more general results that appear in Section 4, where the sides of the holey hexagon HL,R may n,2m approach infinity at different rates (that is, 2m ∼ ξn for some real positive ξ). For ξ (cid:54)= 1 these interactions either blow up or shrink exponentially, thus the results presented above are for the special case where ξ = 1. While this special case agrees completely with Ciucu’s tori result [5], the asymptotic analysis presented in Section 4 shows that in the planar case such an interaction depends somewhat delicately on the rate at which the “edges” of the region approach infinity. A discussion of similar findings may also be found in [9, Remark 1]. The following section establishes exact formulas that count weighted tilings of H(cid:98)L,R n,2m and HqL,R . The asymptotic behaviours of these formulas are established in Section 4. n,2m Throughout both sections the following notation for generalised hypergeometric series is used: (cid:20)a , ..., a (cid:21) (cid:88)∞ (a ) ···(a ) zk 1 p 1 k p k F ;z = , p q b , ..., b (b ) ···(b ) k! 1 q 1 k q k k=0 where (α) is the Pochhammer symbol, defined to be β (cid:40) α·(α+1)···(α+β −1) β > 0, (α) = β 1 β = 0. INTERACTIONS BETWEEN INTERLEAVING HOLES IN A SEA OF UNIT RHOMBI 9 O Figure 8. A tiling of Hq{−2,6},{0,−8} displaying lattice paths across unit 10,4 rhombi, left, and the corresponding set of lattice paths starting at a set of green points and ending at a set of red ones, right. 3. An Exact Formula The main goal of this section is to prove the following enumerative formula. Theorem 3. The number of tilings of HL,R is n,2m (cid:32) (cid:33) n 2m n (cid:89)(cid:89)(cid:89) i+j +k −1 q ·detE ·detE(cid:98) , R,L R,L i+j +k −2 i=1 j=1k=1 q where E and E(cid:98) are p×p matrices defined in Theorems 5 and 7 below. R,L R,L Remark 4. The leftmost product in the above theorem is easily recognisable as MacMa- hon’s much celebrated box formula [18], named so because it counts the number of unit cube representations of plane partitions that fit inside an n × 2m × n box. This is equivalent to the number of rhombus tilings of an un-holey hexagon5 of side lengths n,2m,n,n,2m,n (going clockwise from the southwest edge) and is denoted H 2m. n The above theorem follows from establishing exact formulas that enumerate the (weighted) number of tilings of the regions H(cid:98)L,R and HqL,R . According to Ciucu’s n,2m n,2m factorisation theorem (2.2), the product of these formulas counts the total number of tilings of HL,R . To begin, one translates tilings of these regions into families of lattice n,2m paths across dimers in the usual way6, which in turn correspond to families of non- intersecting lattice paths consisting of north and east steps that begin at a set of points 5A hexagon that contains no holes. 6To generate such a family, start points are set in the centre of the vertical edges of dimers that lie along the west edge of each region, along with the dimers that lie along the vertical edge of each left pointing triangular hole. One then draws a path across dimers by travelling from one side of a dimer to the opposite parallel side, thereby creating a family of non-intersecting paths that corresponds to precisely one tiling. 10 T. GILMORE O Figure 9. A tiling of H(cid:98){−2,6},{0,−8} displaying lattice paths across unit 10,4 rhombi, left, and the corresponding set of lattice paths starting at a set of green points and ending at a set of red ones, right, where the yellow weighted tiles on the left correspond to the circled points the touch the main diagonal on the right. A = (A ,...,A ) and end at a set of points E = (E ,...,E ), where 1 m+p 1 m+p (cid:40) (i,1−i) 1 ≤ i ≤ m, A = i (n + li−m +1, n + li−m) m+1 ≤ i ≤ m+p, 2 2 2 2 and (cid:40) (n+j,n+1−j) 1 ≤ j ≤ m, E = j (n + rj−m +1, n + rj−m) m+1 ≤ j ≤ m+p. 2 2 2 2 Tilings of HqL,R correspond to families of non-intersecting lattice paths that begin at n,2m A and end at E such that no path touches the main diagonal (that is, the line y = x), see Figure 8. The weighted count of tilings of H(cid:98)L,R instead correspond to the weighted n,2m count of families of non-intersecting paths from A to E that do not extend above the main diagonal, and where each path P that touches the main diagonal at T(P) many points has a weight of 2T(P) (see Figure 9). Remark 5. Suppose σ is a permutation on m+p letters that maps each start point A i to an end point E . It should be clear that due to the constraints on the families of σ(i) paths described above, every family of non-intersecting paths arises from precisely one permutation which is uniquely determined by the positioning of the triangular holes relative to each other. According to the well-known theorem of Lindstro¨m [17], Gessel, and Viennot [12], if σ is the only permutation on m+p letters that gives rise to a family of non-intersecting paths that begin at A and end at E, then the number of such (weighted) paths is given by the determinant of the matrix P = (P ) , where P denotes the weighted i,j 1≤i,j≤m+p i,j count of the number of lattice paths from A to E . To be more precise, suppose that i j every vertical or horizontal unit step (edge) between a point a and a point b has weight