- Email: [email protected]
Local statistics of lattice dimers
Ann. Inst. Henri Poincark, Probabilith
Vol. 33, no 5, 1997, p. 591-618.
et Statistiques
Local statistics of lattice dimers Richard KENYON CNRS UMR 128, Ecole Normale Sup&ewe de Lyon. 46, aIlCe d’Italie, 69364 Lyon, France. E-mail: [email protected]
ABSTRACT. - We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy ~1on the space of tilings of the plane with dominos. We construct a measure v on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the v-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for ,LIand Y, and compute the rate of convergence to mixing (the correlation between distant events). For the. measure v we compute the variance of the height function.
Ke)j words: Dimers, dominos %XJM& - Soit p la mesure d’entropie maximale sur l’espace X des pavages du plan par des dominos. On calcule explicitement la mesure des sous-ensembles cylindriques de X. De meme, on construit une mesure v d’entropie maximale sur l’espace X’ des pavages du plan par losanges, et on calcule explicitement la mesure des sous-ensembles cylindriques. Comme application on calcule, pour p et V, les correlations d’evenements distants, ainsi que la v-variance de la fonction “hauteur” sur X’.
1. INTRODUCTION A &mino is a 2 by 1 or a 1 by 2 rectangle, whose vertices have integer coordinates in the plane. Let X be the space of all tilings of the plane with A.M..? Classijication : 82 B 20, 60 J 35. Ann&s de I’lnstirut
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dominos. Then X has a natural topology, where two tilings are close if they agree on a large neighborhood of the origin. In this topology, X is compact, and Z2 acts continuously on X by translations. Burton and Pemantle [BP] proved that there is a unique invariant measure p of maximal entropy for this action. This measure has the following property: let T be any tiling of a finite region R, and U, the set of tilings in X extending T (we call UT the cylinder set generated by T). Then p(Ur) only depends on R and not on the choice T of a tiling of R. In this paper we compute h(Ur) explicitly, for any finite tiling T. We will prove in section 5 that: THEOREM 1. There is a function P: Z2 + C with the following property. Let E be any jinite set of disjoint dominos, covering “black” squares bl, . . . , bk and “white” squares wl, . . . , Wk. Then ,Q(UE) is the absolute value of the determinant of the k x k matrix M = (mij), where m;j = P(bi - wj). (H ere b; - wj is the translation vector from square wj to square b;.) We call P the coupling function for ,u. It can be computed explicitly (see Theorem 16 below), and the values of P are in Q $ :Q. A similar result holds for tilings of the plane with lozenges. A lozenge is a rhombus with side 1, smaller angle 7r/3, and vertices in the lattice Z[e 2*i/3] . We define a measure v on the space X’ of lozenge tilings, which is the limit of uniform measures on periodic tilings, and show that its entropy equals the topological entropy of the Z2-action. (For a background on measure-theoretic and topological entropy see [PI.) We conjecture that v is the unique measure of maximal entropy. We compute the coupling function for v in Theorem 9: it takes values in Q $ $Q. We give several applications of these two results. First, it is known that the translation-action of Z2 on dominos or lozenges is topologically mixing (if two events are far enough apart, one can find a single tiling containing both of them). We show here that it is strongly mixing for the measures p and V, that is, events that are far apart are almost independent. In particular we show that the convergence rate to independence is at least quadratic in the distance, that is, for any two finite tilings Tl and T2, and u E Z2,
and similarly for V; see Theorem 11. There is a corresponding lower bound when Tr, Tz are single tiles. Correlations for single tiles have been previously computed by Fisher and Stephenson [FS] and Stephenson [St]. Annales
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Another application is to the computation of the variance of the height function for v. The height function of a lozenge tiling was introduced by Blote and Hilhorst [BH]. We compute for v the variance of the height function in the y-direction (Theorem 15), that is, the variance in height between two vertices separated by distance 71 in the y-direction. This variance is related to the expected number of “contour lines” separating the two points (Proposition 12). The variance in other directions can also be computed with the same methods, although we could not obtain a closed form. On the other hand, domino tilings also have a height function, but we were not able to compute its variance using the methods here. A third application which we will not discuss here is to the computation of entropy ent(s, t) of tilings whose height function has a fixed average slope, that is, the set of tilings of the plane whose height function ht(.. .) satisfies ht(z, y) = ht(0, 0) + sx + ty + o( 1x1, 1~1).This computation is used in [CKP] to give a method of counting the approximate number of tilings of any region, by maximizing an entropy functional over the region. The results of this paper are very general. Using a result of Kasteleyn [K2], we can compute cylinder set measures (for certain measures of maximal entropy) for the space of perfect matchings on any periodic planar graph. Here “periodic” means the graph is finite modulo an action of 2’ by translations. In particular Hurst and Green [HG] showed that the Ising model on a planar graph with nearest-neighbor interactions can be represented as a matching problem on another planar graph (actually they showed this in an equivalent form). Our method therefore allows one to compute the local probabilities for the Ising model. Indeed, methods similar to those in this paper were used in Montroll, Potts and Ward [MWP] to compute long-range correlations for the Ising model. The coupling function which we define is analogous to the Green’s function for a resistor network (see [BP]). Indeed, subdeterminants of the Green’s function matrix give probabilities of finding subsets of edges in a random spanning tree. There is a well-known connection between domino tilings and spanning trees: Burton and Pemantle [BP] have found, via enumeration of spanning trees, the measures of certain cylinder sets for dominos. Our method, which is different, gives the measure of all cylinder sets and is adaptable to planar graphs where the spanning tree connection is not known to hold. The structure of this paper is as follows. Section 2 discusses background: we show how to compute the number of tilings as a determinant, and we define ,u, v and the height function. In section 3 we show how to compute the cylinder set measures for finite planar graphs. In section 4 we compute Vol. 33. Ilo s-1997.
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explicitly the coupling function for v and give the applications. We chose in this section to concentrate on the case of lozenges rather than dominos since lozenges seem to be less prominent in the mathematics literature. Section 5 restates the results of section 4 for dominos.
2. BACKGROUND 2.1. Matchings
on finite planar graphs
Let G be a graph. Two edges of G are disjoint if they have no vertices in common, A perfect matching of G is a set of disjoint edges which contains all the vertices. If G is finite, connected and bipartite, we say G is baIanced if there are the same number of vertices in each half of the bipartition. Let Z2 denote the graph whose vertices are Z2 and whose edges join every pair of vertices of distance I. It is easy to see that domino tilings of the plane correspond bijectively to perfect matchings of Z2. The graph Z2 is bipartite: color the vertex (x,9) black if z + y E 0 mod 2, otherwise color it white. Note that then a domino “covers” exactly one vertex of each color. Similarly, lozenge tiiings correspond bijectively to perfect matchings of a particular bipartite planar graph H, the “honeycomb” grid. The graph H is the l-skeleton of the periodic tiling of the plane with regular hexagons (see Fig. 1). Rather than use lozenges as defined in the introduction, we
Fig. 1. - A matching of the corresponding
on the graph H and some lozenges (dotted lines).
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adopt the following coordinates. Let 2 = $ - 9 and i = $ + q. The set of vertices of H is V = V, U VI, where VO = {a? + b$ ( a, b E a} are the “black” vertices, and VI = 1 + V. are the “white” vertices. There is an edge between every pair of vertices at unit distance. We will denote the vertex a? + bi by the triple (a, b, 0), and a vertex ai + b$ + 1 by the triple (a, 6, 1). Hopefully denoting vertices with ordered triples will not confuse the reader. Note also that a lozenge has side & in these coordinates. Kasteleyn [Kl] and Temperley and Fisher [TF] independently computed the number of perfect matchings of an m x R. grid (in other words, the number of domino tilings of an m x n rectangle). Kasteleyn in [Kl] also computed the number of matchings on a toroidal graph in the case of dominos. Later, in [K2], he showed how to compute the number of perfect matchings of general planar graphs (thereby including the case of lozenges). His method is to count tilings using Pfaffians. We outline here an adaptation of his proof for the case of a simply-connected subgraph of H. We say that a subgraph of H is simply connected if it is the l-skeleton of a simply connected union of “basic hexagons” of H. The following is a special case of the results of [K2]: THEOREM 2. - Let H’ be a Jinite simply connected subgraph of H. The number of pe$ect matchings of H’ equals dm, where A is the adjacency matrix of H’.
Here is a sketch of the proof: since the graph H’ = (V’. 23’) is bipartite, we have V’ = Vd U V,’ (where Vi c V,, Vi c VI) and E’ c Vd x V{. Order the vertices V’ so that those in Vi comes first. We then can write
where B cardinality matchings matrix B
: RlcGl --+ BBI”:l. We can assume that Vd and V{ have the same n (that is, H’ is balanced), for otherwise there are no perfect (and det (A) is clearly 0). Then the determinant of the square is
(2.2)
detB=c sgn(~)a,(l) . .G,~(~~,)
where A = (aij) and the sum is over bijections c vr/;r+ Vi. Each nonzero term in the sum corresponds to a matching. One need only check (and this is the most interesting part, although we won’t do it here) that each term has the same sign. Thus 1det(B)I = dm = # of matchings. Vol. 33, no 5-1‘397.
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This same method also works for dominos, except that one must modify the adjacency matrix to make the signs in (2.2) work out right. One way to modify it is to put weight i = fl on vertical edges [WI. It is slightly more complicated to count matchings on a graph embedded on a torus. Kasteleyn showed that, for a graph on a torus, one can count matchings using a sum of 4 Pfaffians (see Lemma 7 below). 2.2. The measure of maximal entropy An important issue not dealt with by Kasteleyn is the appropriate sense in which random tilings of large finite regions in the plane resemble random tilings of the whole plane. Indeed, as the work of [EKLP] shows, the local configuration of the boundary of a region has a drastic effect on the number of perfect matchings, and hence one must take care when computing the measure p as a limit of measures on tilings of finite regions. A tiling with free boundary conditions, or simply free tiling, of a region R is a tiling of R, where the tiles are allowed to protrude beyond the boundary of R. The topological entropy of domino tilings is by definition
where 5’1, is the set of free tilings of a k x Ic square. Here the square regions Sk can be replaced by any sequence of “sufficiently nice” regions, for example, convex regions containing squares of size tending to infinity. In this case one divides by the area of the k-th region, not by k2. The same definition also applies to lozenge tilings (replace the k x Ic square with for example the set of vertices in a k x Ic square centered at the origin). The entropy H(,u’) of a translation-invariant tilings of the plane can be defined by
measure 11’ on domino
We have H(p’) 5 Htop for any translation-invariant measure $ on X, since, as the reader may readily show, for a finite set of real numbers {ui} satisfying ai > 0 and C ai = 1, the quantity - C ai log ai is maximized when the ai are equal. Andes
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For any measure p’ defined on the finite set X(S) region S, the entropy (per site) of $ is
of tilings of a finite
where /l’(s) is the probability of the tiling s. 2.2.1. The domino measure. - Burton and Pemantle showed: THEOREM 3 ([BP]). - There is a unique measure of maximal entropy p for domino tilings of the plane. The entropy of p equals the topological entropy J&op(X).
Let Zk,, be the quotient of Z2 by the lattice of translations generated by (am, 0) and (0,271). Then Z&,, is a graph (on a torus) with 4mn vertices. Let CL~,~ denote the uniform measure on perfect matchings of Zk,,. Let ,u* be a weak limit of pm,n as m, n -+ 00. We claim that p* = ~1. To see this, Kasteleyn [Kl] showed that the entropy of pL,,, converges as m, n -+ cc to the entropy of CL. So it suffices to show the limit of the entropies is the entropy of the limit. Let Wk by a k-by-k window in Z2 (or on ZL,, when m and n are larger than k). Let pL,,,(Wk) denote the restriction (projection) of pm,n to Wk. Then p,,,(Wk) converges to ,u* (Wk) by weak convergence. Since there are only a finite number of configurations on Wk, lim H(i-hn(Wk)) nL,R’cc
Since ff(~h,~(Wk))
2 H(p.,,,,)
= ff(~*(Wk)).
we have
Taking limits over Ic gives H(p*) = Htop and therefore by the theorem p* = p. We would like to thank Jeff Steif and Jim Propp for the idea behind this argument. 2.2.2. The lozenge measure. - Burton and Pemantle’s proof of Theorem 3 above uses a connection between spanning trees and domino tilings. Lozenge tilings are in fact also in bijection with a set of spanning trees, or rather, directed spanning trees (“,arborescences”) on a directed triangular grid, see [PW]. However it is not evident how to adapt the proof of [BP] to this case. We will show nevertheless that the uniform measures on certain Vol. 33. no S-1997.
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toroidal graphs converge to a measure I/, and their entropy converges to the topological entropy. By the argument following Theorem 3, the entropy of 11 equals the topological entropy. The only fact missing is the uniqueness of Y. be the quotient of H by the lattice of translations generated Let &,, by rni and Qj. Then H,,, is a graph with 2mn vertices. Let II,,,, be the uniform measure on matchings of H,,,,. LEMMA 4. - The entropy of u,,,, converges to the topological lozenge tilings of the plane.
entropy
qf
Proof - The proof is a simple variant of an unpublished trick due to G. Kuperberg. Let F, be the set of free lozenge tilings of the region R,, where R, consists of an equilateral triangle of side nfi (recall that a lozenge has side length a). Then F, is partitioned into equivalence classes, two tilings being equivalent if they have the same set of tiles which protrude beyond the boundary. The number of equivalence classes is less than C” for some constant C, so some equivalence class B, must satisfy
Reflect the region R, along one of its edges c. Given any two tilings 7’1, T2 E B,, there is a tiling of the union of R, and its reflection, where the R, is filled with Tl and its reflection is filled with the reflection of Tz. This is because the tiles which protrude across the boundary edge c are fixed under this reflection. Take the union of 18 reflections of R, so as to make the parallelogram of Figure 2. Given any 18 elements of B,, we can fill them in the 18 triangles
Fig. 2. - The 18 triangles make a torus. Ann&s
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to make a tiling of this whole parallelogram. Furthermore, the configurations of tiles protruding beyond opposite boundaries of this parallelogram are translates of each other. As a consequence we can glue opposite sides of the parallelogram together to make a tiling of the 3n by 3r1 torus (tilings of which correspond to perfect matchings of H3n,3n). We therefore have (where N,,, is the set of tilings of Han,zl,)
This implies that
which completes the proof. q We will see later that H(v,,,) and N(v,,,) have the same limit as m, n + cx), so the restriction to the tori E&, is unimportant. In fact we will show in section 4.4 that there is a unique weak limit to the v,,,!. Let 11 be this limit. We make the following CONJECTURE. - The measure v is the unique measure ,for lozenge tilings of the plane.
of maximal entropy
2.3. Height fmctions For any perfect matching of a simply connected subgraph H’ of H one can associate an integer-valued height fwwtim to the faces of H’. (By face of H’ we mean a basic hexagon which contains an edge of H’.) Let T be a perfect matching of H’. Pick a face zu arbitrarily and assign it height h(zO) = 0. For any other face z, take a path x0, x1, . . . ,x, = x of faces, where for each 3, zj+l is adjacent to ‘cj along an unmatched edge. Let UI = e2xi/3. Then h(~j+~) = h(zj) f 1, where h increases by 1 if ~;j+l - zj is in directions i, iw, or iw2 and h decreases by 1 if zj+l - :~j is in directions -‘i, -iw or -iw2. This defines a height on each face, which can easily be shown to be independent of the choice of path to that face. Figue 3 shows a height function for the matching in Figure 1. For dominos there is a similar definition; see [T]. For a simply connected subgraph H’ which has a perfect matching, the height function along the faces bounding the region (that is, faces outside H’ but containing an edge of H’) is well’defined up to an additive constant and is independent of the matching [T] (rather, this is true if the union of H and these faces is still simply connected). Vul. 33, Ilo 5.1997.
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Fig. 3. - Height
3. THE MEASURE
function
for lozenges.
OF A CYLINDER
SET
The results in this section are stated for lozenges, but they apply verbatim for dominos, on condition that one puts weight i = fl on vertical edges in the adjacency matrix B. Let H’ be a finite, balanced, simply connected subgraph of H, with vertices {q , . . . ,w,} E VF and {vi,.. . , w;} E VI. Let E = {ei,. . . ,ek} be a subset of disjoint edges of H’, with ej = wP,vi3 for j = 1, . . . , k. In the formula (2.2), each nonzero term in the sum corresponds to a perfect matching; those terms whose matchings contain all the edges in E are precisely the terms which contain the subproduct
The sum of the terms containing this product is (up to sign) the product of aE with the determinant of the cofactor BE, that is, the determinant of the matrix obtained from B by removing all rows pl, . . . ,pk and all columns qi,..., qk. This can be proved inductively as follows. Expand det(B) along the row pl: det(B)
= (-l)pl+l
(apI1 det Bpl,l - up12 det Bpl,2 +. . . fapln Ann&s
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The only term that contains aplql is the term for column 41, which is the term (-l)pl+qlapIql det BpIql. Now expand BPIql along row pz, and so on. Thus the sum of the terms containing aE is (-1)x l-‘J+qJaE det( BE). Since laE/ = 1 this proves PROPOSITION 5. -
The number of peeect matchings containing all edges
in E is
Later on the sign of det (BE) will be important to us. We note therefore that the number of perfect matchings containing all the edges in E is (-l)~(p~+y~)a~
det(Bs)sign(det
B).
In a more general setting (e.g. in [CKP]) one might wish to consider weighting the edges non-trivially, in which case the formula (aE det(BE)I counts each matching according to the product of the weights on its edges. Note that the quantity det(BE) does not depend on the actual set of edges in E, but only on the set of vertices involved. This shows, as expected, that each matching of the vertices of E can be extended to the same number of perfect matchings. The matrix BE is an (n - k) x (n - k) cofactor of B; its determinant is equal to det( B) times the determinant of the k x k cofactor of the inverse of B, (B-~)E-, where E” is the set of rows and columns not involved in E. So we have THEOREM 6. -
The number of per$ect matchings containing all edges
in E is 1det((B-l)p)
det(B)I.
That is, I*ff’(UE)
=
1 det((Bvl)E*)I.
More precisely, PHI
= (-l)~pJ+qJaEdet((Bsl)E*).
The advantage of this over Proposition 5 is that the computation is a determinant of size k rather than R. - k. In particular if E consists of a single edge V~IJ~,the probability that that edge is in a random matching is the absolute value of the ij-th entry of B-l. The rest of the paper consists of the computation of B-l and applications of Theorem 6. Vol. 33, no 5-1997.
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4. LOZENGE
TILINGS
4.1. The torus Recall that IS,,,, is the toroidal graph H/(mZd + n@), and the vertex r? + si f t is designated by the triple (T, s, t) E Z/mZ x iI/& x (0, l}. Let Al be the adjacency matrix of H,,,,. Let A2 be the matrix obtained from Al by changing the sign of the entries corresponding to edges from (m - 1, k, 1) to (0, k, 0) f or each k E Z/G!. Let A3 be the matrix obtained from Al by changing the sign of the entries corresponding to edges from (k, 78- 1,l) to (k, 0,O) for each k E Z/ma. Let A4 be the matrix obtained from Al by changing the sign of both these sets of entries. Let Br , &, Bs, BJ be the matrices obtained from Al, AZ, AS, A4, respectively, as in (2.1). Following Kasteleyn, on condition that m, n are both even, we have LEMMA
(4.1)
7. -
Z,,,
The number of perjkt
= f(-
matchings Z,,,
of H,,,
is given by
det B1 + det Bz + det Bs + det Bd).
A term for each perfect matching appears in each of these four determinants. The signs are arranged so that each perfect matching is counted three times with sign + and once with sign - (see [Kl]). For other parities of m and n a similar equation holds but with different signs. Throughout the rest of this section we will assume m and 7~ are both even and neither is divisible by 3. This is simply a matter of convenience; the computations go through in the other cases but require slightly different arguments. Theorem 6 also applies to tilings of the torus. The quantity det B1 det(B,l)E* contains a term for each matching containing ail edges of E. The sign of these terms is the same as the sign of the corresponding terms in det(&) (this easily follows from the proof). This is also true for Bz, Bs, &, and so LEMMA 8. - The number ofpeflect matchings o~T&~ containing a given set of disjoint edges E is given by
(4.2)
f/ - det B1 det(B,t)E.
+ det B2 det(B,l)E*
+ det Ba det(B;l)E*
+ det B4 det(B,r)E*
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4.2. Computation
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of the eigensystem
The eigenvectors of the Aj are functions f(r, s, t) = C&WY, and f’(r. s, t) = C:Z”W where co = dl + z-1 + W-l Cl&m CL = co c; = -cl (for either choice of sign of the square root), and
The corresponding
zm = 1, ,p zz -1,
wn = 1 wn = 1
for Al for AZ
.P = 1, ,p = -1,
w”=-1 wn=-1
forA forA4.
eigenvalues are
x = COCl X’= -c()q. Consider for example the matrix Al . Let
Ck,P,l
=
1 + 2$ + UI:
c:,,,o = clc,e.o I (any choice of sign for the square roots will do). Then fk,e(r, s, t) = Ck,e,&w: and &(T, s,t) = c~,~,~z~w:S are eigenfunctions. The functions fk,e + fL,e are zero on VI, and run through a basis for all Vol. 33, no s-1997
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functions on V, (the exponentials) as (k:, !) runs through (0, . . , m - l} x (0,. . . , n. - l}. Similarly the functions fi~,~ - fL,[ are zero on V. and run through a basis for functions on VI. Thus the collection of fk,e and fL,y is a basis of eigenvectors for the matrices Al. In particular we have (4.3) det(Al)
=
=
n
l-j
k=O
e=o
m-l
n-l
k=O
e=o
m-l n
n-l n
k=O
e=o
Ak,eA&e
(1 +
,$W+n
+
e27ee/n)2
(we removed the initial minus sign in this last equality since there are an even number of terms, and also in the second product we replaced k, /2 with -k, 4). Similarly one can compute (4.4) det(A2)
= n(l
+ ,pw+l)h
+ e2wn)(l
+ e-4wl)/m
+
e-27wn)
k,e =
l3 Ic.e
l+e
?ri(2k+l)/m
+
e2rie/n
2 >
and furthermore (4.5) (4.6)
det(A3) = n(l k,e det(A4) = n(l k.e
+ $niklm
+ pW+l)ln)2
+ exi(2k+l)/m + e42e+l)/n)2,
These four determinants are non-zero by our assumption that m and n are nonzero modulo 3 (three complex numbers of modulus 1 sum to zero only if they are a rotation of the set { 1, e2ri/3, e4?r’/3}). 4.3. Computation
of the inverses of I&, . . . , B4
It suffices to invert the A?. Define the vector fio,o,o(?Y7t)
1 if(z y t)=(O,O,O) = { o othe;w;se Ann&s
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Consider first the matrix Al. We can write
~o,o,o(& Y>t> = &
1
fk,e(? Y(/, t> + G,eh Y>t) ck,e,o
k.t
Thus the vector
has the property that
&Po,o,o(TY, t>
= SO,O,O(G YV,t).
The preimages of delta functions at other vertices can be similarly defined: define Pr+,o (z,y,t) = PO,O,O(X - r,y - s,t); then
Thus the function PT,S,o gives the entries of the (r. s, 0)-column of the matrix AlI. For notational simplicity we will write P(l)(z, y! t) = Po,o,o(z, y, t) (the superscript (1) refers to the fact that it comes from matrix A,; note that I’(‘) also depends on m and n). If we simplify (4.7) above: .fk,&,
y,t) - G:,P(~>Y>t) >
xk.P
,
we see that P(l) (z, y, 0) = 0 for all 2, Y. On the other hand plugging in for f, c, A, zl, wr we have (4% Similar expressions hold for the inverses of AZ: As, Aa, except that the sum is over a different set of angles. For example for A2 we have eKi(2k+1)~JlrLe2~iVy/n
P(2)(z, y, 1) = +& ck Vol. 33, n” 5.1997
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4.4. The limiting
measure
Here we compute the limit of the ratio of (4.2) and (4.1), that is, the limit of the cylinder set measures v,,,( UT). We still assume m, n $ 0 mod 3 so that the determinants in (4.3)-(4.6) are nonzero. We will show below that the quantities det((BJT1),!+) forj = 1,2,3,4 all converge to the same value as m, 71 -+ co. Then (if we remove the absolute value sign in (4.2)) the ratio of (4.2) and (4.1) is a weighted average of these four quantities, with weights f det Bj/Z,,,,. These weights are all in the interval [-1, l] since Z,,, 2 (det By/ for each j: recall that Z,,,, and 1det Bj ( count the same objects except that some signs in 1det Bj/ are negative. Since the weights sum to 1, the weighted average is also converges to the same value as each det( ( BJT1)p ). It suffices to show that, for fixed X, y, for each j = 1,2,3,4 the quantities P(~)(x, g/,1) tend to the same value (recall that the matrix BJ” has entries which are the values of P(j)). For a fixed integers 5, y, the function
is continuous on ([O, 2n] x [0,27r]) \ { (27r/3,47r/3) U (47r/3,2~/3)}, and has a pole at the two points (2~/3,47r/3) and (47r/3,2n/3). The expression (4.8) and the corresponding expressions for the other P(j) are each Riemann sums for the integral of p over [0,2~] x [0,27r]. Fix 6 > 0 and let N6 be a 6 x a-neighborhood of the pole (&, &) = (2~/3,4x/3). Outside N6 (and the corresponding neighborhood of the other poie) the Riemann sums converge to the appropriate integral. We must show that on NS the sums are small (in the sense that they tend to zero with S as m, n --f co,). On Ns, we use the Taylor expansion of the denominator:
(@,d)ENh ILC mn
ZZ-
N6 1 - ieuieo(B
1
1
mn cN6 Ie-2pi/3(8
=-
- t90) -
- 00) + e-4?ri/3(4
- tie) + O((0
1 mn
cNg
(,-21ri/3(8
_
0,)
:
e-47ri/3(t#,
Annales de l’lnstitut
_
40)l
- 60)2, (t#~ - c$~)~)]
1 +
Hem-i PoincarP
‘(’
-
go> 4 -
- ProbabilitCs
$O)-
et Statistiques
607
LOCAL STATISTICS OF LATTICE DlMERS
(The reason we can take the big-0 term out of the denominator is that (,-2Ti/3(0
-Oo)+
e-4d3
(4 - 4011 2 const(lo
- 001 + 14 - 401)
for 10 - 601 and 14 - q50 ) small; we can then move it out of the summation altogether because of the factor &.) Since m, n f 0 mod 3, we have, in the sum for each P(j), that 8 - Ho is of the form 8 - B. = r( 2 - 3) = 7r& and similarly q5- & = 5, where 5’ and e’ are not multiples of 3. Thus the sums are each bounded by one of the form -1
mn c
where the sum is over
This sum is easily shown to be bounded independently of m, n and tending to zero as 6 ---t 0: this follows from the integrability near the origin of the functh ,r+eAi3y, with respect to f&&y. THEOREM 9. - For ajnite matching E covering black vertices { bl , . . . . bk } and white vertices { wl? . . . , w~c}, we have v(UE) = 1[let, P(w; - bj)], where 2n P(x,
2m
y; 1) = -& .o Is
0
ei"8eiy&dedq 1 + e-i8 +
e-im'
Pro06 - This follows from Theorem 6 and the convergence argument above. 0 4.5. The coupling function
for lozenge tilings of the plane
We call P(x, y, t) the coupling function for lozenge tilings of the plane. Let .z = ePie and w = ePid. Then we obtain the contour integral
The function P has all the symmetries of the graph H, that is, (4.10)
P(x,y,l) = P(y,x,l)
Vol. 33. Ilo s-1997.
= P(-z
- y - l,z, 1) = P(y,-3:
- 9 - 1,l)
= P(-,
- y - l>?/,l)
- y - 1,l).
= P(:c,--3:
608
R. KENYON
These are obtained by noticing that (4.9) is invariant under interchanging z and w, or under the substitution (z,ru) -+ (w-l, ZW-l), or under combinations of these. We can explicitly evaluate the integral (4.9) as follows. This is slightly easier to calculate in the case z _< - 1. The other values can then be obtained by (4.10). If we fix w and integrate over Z, there is a pole inside the unit circle when 11 + WI < 1, that is, when -q5 E (2~/3,4~/3). The residue of z-“-r/(1 + z + w) at this pole is (-1 - w)+-l. For -q5 6 (2~/3,4~/3) there is no pole inside the circle and therefore the integral is zero. So when x 5 -1, e4*t,3
(4.11)
P(x, y, 1) = 2(-l)-“-’
s &2TZ/3
W
-y-y1
+ w)-“%w.
For example when x = -1 we have (pyp
(4.12)
_
e4TTiy/3
Y
>
- 4 27ry ’
where cy = 0, 1, -1 if y E 0, -1,1 mod 3. Some values of P(x, y, 1) are plotted in Figure 4. To compute these values, start from (4.12), which, along with the symmetries (4.10) gives the value on 3 lines, and use the
Fig. 4. - Values
of P(z, Andes
y, t). Here de l’lnstitut
T = &/(h).
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fact that the sum of the three values adjacent to any vertex (except the origin) is zero (since AP = 0 at these vertices). For example the equation P(-k,O?
1) + P(-k
determines P(-k, 0,l) before that P(-k,-1;l)
+ 1, -1: 1) + P(-k
+ 1.0.1) = 0
for k > 0 inductively for increasing k (note as = P(-l,-k,l) = $).
4.6. Some local probabilities
for lozenges
From Theorem 6 and equation (4.11) with z = - 1, y = 0, the probability of the edge (0, 0,O) (0, 0,l) being in a random matching is given by P(0, 0,l) = $, as expected by symmetry. The probability of both the edges (O,O, O)(O, 0,l) and (0, l,O)(O, 1,l) occurring is given by
Po,o,o(O, 0, 1) ~0,0,0(07 171) PO.l,O(O,
0, 1)
The probability
~0,1,0(07
171)
z 13%. =
of the three edges
CO,% O)(O,(Al), ((Al, ‘4-L 1;1) and(-l, l, 0)(-l, ($1) (which bound a hexagon) all occurring, is:
(Such a hexagon corresponds to a local minimum for the height function.) Note from (4.12) that P(-3k,3k, 1) = P(-1, -3k, 1) = 0. As a consequence, the set of edges E = {en}nEZ where e, is the edge (-3n, 3n, O)(-3n, 3n, 1) is an “independent” set: the probability of any fixed subset of size k occurring in a random matching is exactly ( 1/3)L. Is there any simple explanation for this fact? We can also compute long-range correlations. The probability of the horizontal edge (0, 0, O)(O, 0,l) and the horizontal edge (-71,n, 0)(-n, n, 1) both occurring is given by the determinant 1 5
P(n, Vol. 33, no 5.1997.
-n,
P(
1)
-n,
n,
1 3
1) =
f
-q-71,72,1)”
610
R. KENYON
By (4.10) and (4.12), P(- ?L,71, 1) = P(-1, -n, 1) = I,&/. the probability of these two edges both occurring tends quadratically not faster) to i. More generally we have PROPOSITION
so
(but
10. P(x, y, 1) = 0
( I4 : IYI > . Proof. - We show for z 5 -1 that P(z, y, t) = 0( 6); by the dihedral 6-fold symmetry of P the result will then follow. From (4.11) we have (1+ w(-+-ildW/ =-
2 2r
?r
(2 cos(e/2))-z-‘de
.I &r/3
0 where we used the substitution t = 2cos(0/2). Let Ei and Ea be two finite collections of edges. The probability edges El U Ez are present in a matching is given by a determinant
that
(4.13)
where det(C1) = ~(UE~) and det(C,) = p(U,,). The entries of D1 and D2 are the values of P connecting points of El to Ea. If the distance between El and E2 is 1 n then these entries are all O(i). Let c +J)~l,(l) . a. uka(k) be the expansion of the determinant (4.13). Suppose aie(~) is an entry from the submatrix Di, that is, i 5 IElI and c(i) > J&I. Then there must exist an index j with j > [El/ and c(j) 5 JEil (since c is a bijection), that is, ajOcj) is an entry of Da. We may similarly conclude that each term in the sum for the determinant has the same number of entries from Di as from Dz. Therefore the determinant (4.13) is equal to det(Ci) det(C2) + 0( 3). This proves THEOREM
11. - If the distance between El and E2 is at least n, then
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Note that the comments before Proposition (10) give a corresponding quadratic lower bound if El and E:! are single horizontal tiles separated by -(3n + l)? + (3n + l)$ An argument similar to Theorem 11 shows that v is mixing of aII orders, that is, given k events El,. . . , & each of which is separated from the others, the joint probability converges to the product of the individual probabilities as the distance between them tends to infinity. 4.7. The height distribution Let h, be the random variable which gives the height difference between the face at the origin (by which we mean, the face above and right of the point (0, 0,O)) and the face at the point C--r&,n, 0) in a random lozenge tiling of the plane. In this section we compute the variance of h,. Since the height is only defined up to an additive constant, we may as well assume that the height on the face at the origin is zero. One motivation for computing this variance is that it is related to the number of cycles in the union of two random matchings as follows. Given two random matchings Mi and Mz, their union is a set of disjoint cycles. The difference of their height functions h (l) - hc2) has the property that it is constant on the connected components of the complement of these cycles, and changes by 3 when crossing any of these cycles. The height difference across a cycle increases or decreases by 3 depending on which of the two possible matchings of the cycle occurs in M1 (with the complementary matching necessarily occurring in Mz). Since the two matchings were chosen randomly, across each cycle the difference of heights increases or decreases by 3 independently of what happens at the other cycles. (Among all pairs of matchings whose union has this particular cycle structure, each matching is equally likely to have either “half’ of each cycle.) Thus for a face f separated from the face at the origin by Ic cycles, the variance in the height difference (that is, the variance g2(Ah(f)) = a2(h,(1)(f)-h(2)(f))) is th e same as the variance of a sum of k independent fair coin flips of values f3. This variance is 9k. Since n/l, and IV2 were chosen randomly, the number y of cycles separating f from the origin is also a random variable. We have a2(Ahly) = 9y, and
“2(Ahly)= WW21v>- Whld2 = E(W)“ld since E(Ahjy) a2(Ah)
= 0. Furthermore,
= E((AIL)~)
= c
P(y)E((Ah)21y) Y
Vol. 33, no 5-1997
= c
P(y)9y Y
= 9E(y).
612
R. KENYON
Thus in general the variance of the height differences at a fixed face (for two matchings) is 9 times the expected number of cycles separating that face from the origin. Since the two matchings MI, it& are chosen independently, the variance of their height difference at f is exactly twice the variance of the height of a single matching at f (note that the height at f has mean value zero, which follows from the fact that the probability of any edge being present is l/3). We conclude 12. - In the union of two random matchings, the expected number of cycles separating two faces fi, f2 is 2s/9, where s is the variance in the height (between fi and fi) of a single random matching. Let E, denote the set of horizontal edges (-k, k, 0)(--k, k, 1) for k = l,...,n. PROPOSITION
LEMMA
13. - The value of h, is equal to n - 3r,, where r, is the number edges from E; in the matching.
of horizontal
Proof. - This follows from the definition of the height function. On the vertical path of faces from (0, 0,O) to (-n, n, 0), the height increases by 1 for edge of E, not present in the matching, and decreases by 2 for every edge present in the matching. So the height difference is (n - T,) - 2r,. 0 Let A4 = M, be the n x n matrix A4 = (mk,j) where
Then as we saw, the determinant of A4 is the probability that all the edges from E, are present in a random matching. By symmetry (4.10), when e > 0,
zq-e,e, 1) = q-1, -e, 1) = =. For example when n = 5 we have (with r = $)
For each subset S c I$, of cardinality k, let MS be the submatrix of M whose rows and columns are those of M indexed by 5’. Then
LOCAL
STATISTICS
OF LATTICE
613
DIMERS
C-G
p3+qj det(Ms) is the probability that the edges corresponding to S are present in a random tiling, where the sum is over S. Since pj = 4.j in our ordering, the probability in question is exactly det (MS). Let p(z) = i? - cQ.P--l + Q2Zn-2 - . . . + (-l)?lQI, be the characteristic polynomial of M. Then CXI,is the sum over all subsets S c E, of size k of det MS, that is, the sum over S of the probability that S is in the random matkhing. that there are exactly Let d2) = cizo Pk zk where ,& is the probability k edges from E, present. A formula of Ch. Jordan [C] relates the (kk t0 the p,: @k=ak-
(‘;‘)ak+l+
(‘;1)(2k+z-....
From this we can easily derive the following: THEOREM
14. -
We have q(z) = (1-
z)“p
&
. q
(
>
Now if p(z) = n~=“=,(z - Xi), then q(z) = fi(l
- x; + X+,z).
i=l
The expected value of T, is c k/& = q’( 1). we find
so q’( 1) = q( 1) c
Xi = c
X; = trace(M)
= n/3.
This is as expected since the probability of each edge (-k, k, 0)(-k, being present is l/3. Similarly, the variance of T, is given by 2 C2(&)
=
xk2&
-
(ck/&)2
=
q'(1)
+
q"(1)
-
f 0
Vol. 33. no 5-1997.
.
k, 1)
614 We have
R. KENYON
d’(4=d4ci#Li (1 - A; + X$)(1A;$- xj + X,2)’
and so q”(1) = C
XiXj = trace(M)2
- trace(M2)
= (n/3)2 - trace(M2).
i#j To compute trace(M2), 1 < i < n is:
note that M is symmetric and the ith row for
The inner product of this row with itself is
. Similar sums (with y replaced with l/3) hold for the rows i = 1 and i = n. Summing these up for i = 1, . . . , n yields (4.14)
trace(M2)
= % + r2 2(n - 1) + 29
+2(n-4 ___
+
2-I -+qn-
42
I)2
Note that
Ann&s
de l’hsfitur
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Poincarti
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et Statistiques
LOCAL STATISTICS OF LATTICE DIMERS
615
and similarly
l+;+;+...+rI4 =(I+;+;+...+;>-; (1+;+;+...+IA > = log(r) + O(1) - $log(r/3)
+ 00))
= i log(r) + O(1). Plugging this into (4.14) gives trace(M2)
hd4
= t - 7
+ O(l),
which yields
02(r,) = y
+ O(1).
Finally we find THEOREM 15. - Let h, be the height diference between the origin and the point (-n, n, 0). The expected value of h, is 0. The variance in h, is given by a2(h,) = v + O(f). The only previously known result in this vein is in {CEP], where Cohn, Elkies, and Propp found a 0(&k) upper bound for the height variance on a bounded region.
Proof of Theorem 1. - The proof follows from Theorem 6, the fact that ,h,n --+ JL {see the comments after Theorem 3), and a convergence argument analogous to that preceding Theorem 9. 0 In fact a computation analogous to the case of lozenges yields: THEOREM 16. - The coupling function for dominos is 2rr IS0 vol. 33. no 5-1997.
2x 0
e+@+Y'd')&jd+
2 cos(0) + 2i cost
616
R. KENYON
This statement depends on our choice of weighting the vertical edges of Z2 with weight i. As for lozenges, I’(,, y) is zero for black vertices (i.e. when x + y E 0 mod2). Some values are shown in Figure 5 (where the origin is in the lower left comer).
-it 13 w-w 10 4 A )
i 1 1 ; i.4 4 “(4 -..-...-..;...---..------;------------.-. -;) :O . . . . . . . . . . . . . :0 . . . . . . . . . . . . ..-.... :-I( 2+ rr ) 0
Fig. 5. - Values of P(z, y) for dominos.
One can explicitly evaluate, for 2 2 1, P(2z+1,2s)=(-1)”
;-;
l-;+;-...+(-qZ+r--&
[
)
(
>I
and P(2x, 22 - 1) = -iP(22
+ 1,2X).
From these values (and the diagonal symmetry) the others can be obtained inductively using AP = 0 except at the origin. Ann&s
de I’lnstirut
Henri Poincarc!
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et Statistiques
LOCAL
STATISTICS
OF LA’ITICE
As an example, the probability (0, l)(l, 1) both occurring is
617
of the two edges (0, 0)( 1,O) and
I1 =i I ”4.4
DIMERS
4f
=-,
81
The probability of the two edges (O,O)(l,O) occurring is given by
and (0, l)(O, 2) both
This probability was also computed in [BP], although their stated value is incorrect (they have acknowledged the mistake in their arithmetic). Proposition 10 and Theorem 11 also apply to dominos. However the computation of the variance in the height function cannot be done within this framework since there is no simple analogue of Lemma 13.
ACKNOWLEDGEMENT
I would like to thank Oded Schramm, Jeff Steif, Jim Propp and the referee for many helpful suggestions regarding this paper.
REFERENCES IW
H.
[BP1
R.
Kl WPI
L. H.
uw WW
H. N.
IFSI
M.
WGI
C.
[Kll
P.
IK21
P.
W. J. BL~TE and H. J. HILHORST. Roughening transitions and the zerotemperature triangular Ising antiferromagnet, 1. Phys. A Vol. 15, 1982, L631-L637. BURTON and R. PEMANTLE, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer impedances, Ann. Probnb., Vol. 21, 1993, pp. 1329-1371. QMTET, Advanced combinatorics, D. Reidel Publishing Company, Dordrecht. COHN and N. ELKIES, J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J., Vol. 85, 1996, pp. 117-166. COHN and R. KENYON, J. Propp, in preparation. ELKIES, G. KUPER~ERG and M. LARSEN, J. F’ropp,Alternating sign matrices and domino tilings. J. Alg. Combinarorics, Vol. 1, i992, pp. 1 Ii-132. E. FISHER and J. STEPHENSON, Statistical mechanics of dimers on a olane iattice II. Dimer correlations and monomers, f/z~s. Rev., Vol. 132, 1963, ip. 141 I-1431. A. HURST and H. S. GREEN, New solution of the Ising problem for a rectangular lattice, J. Chem. Phys., Vol. 33, 1960, pp. 1059-1062. W. KASTELEYN, The statistics of dimers on a lattice, Physica, Vol. 27, 1961, pp. 1209-1225. W. KASTELEYN , Graph theory and crystal physics, in Graph Theory and Theoretical Physics, Academic Press, London 1967.
Vol. 33, no S-1997
618 W’WI
R. KENYON E. W. MONTROLL, R. B. Ports and J. C. WARD, Correlations and spontaneous magnetization of the two-dimensional Ising model, J. Math. Phys., Vol. 4, 1963, p. 308-322. K. PETERSEN, Ergodic theory, Cambridge studes in advanced mathematics 2, Cambridge university press, Cambridge, 1983. J. PROPP and D. WILSON, in preparation. J. STEPHENSON, king-model spin correlations on the triangular lattice, J. Math. Phys., Vol. 5, 1964, pp. 1009-1024. W. THURSTON , Conway’s tiling groups, Am. Math. Monthly, Vol. 97, 1990, pp. 757-773. H. TEMPLERLEY and M. FISHER, The dimer problem in statistical mechanics - an exact result. Phil Mug., Vol. 6, 1961, pp. 1061-1063. T. Wu, Dimers on rectangular lattices, J. Math. Phys, Vol. 3, 1962, pp. 1265-1266. (Manuscript
received April Revised FeQruaq
22, 1996; 3, 1997.)
Vol. 33, no 5, 1997, p. 591-618.
et Statistiques
Local statistics of lattice dimers Richard KENYON CNRS UMR 128, Ecole Normale Sup&ewe de Lyon. 46, aIlCe d’Italie, 69364 Lyon, France. E-mail: [email protected]
ABSTRACT. - We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy ~1on the space of tilings of the plane with dominos. We construct a measure v on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the v-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for ,LIand Y, and compute the rate of convergence to mixing (the correlation between distant events). For the. measure v we compute the variance of the height function.
Ke)j words: Dimers, dominos %XJM& - Soit p la mesure d’entropie maximale sur l’espace X des pavages du plan par des dominos. On calcule explicitement la mesure des sous-ensembles cylindriques de X. De meme, on construit une mesure v d’entropie maximale sur l’espace X’ des pavages du plan par losanges, et on calcule explicitement la mesure des sous-ensembles cylindriques. Comme application on calcule, pour p et V, les correlations d’evenements distants, ainsi que la v-variance de la fonction “hauteur” sur X’.
1. INTRODUCTION A &mino is a 2 by 1 or a 1 by 2 rectangle, whose vertices have integer coordinates in the plane. Let X be the space of all tilings of the plane with A.M..? Classijication : 82 B 20, 60 J 35. Ann&s de I’lnstirut
Hewn’ Poincari - F%ob&ilit& et Statistiques Vol. 33/97/M/$ 7.00/O Gauthier-Villm
- 0246-0203
592
R. KENYON
dominos. Then X has a natural topology, where two tilings are close if they agree on a large neighborhood of the origin. In this topology, X is compact, and Z2 acts continuously on X by translations. Burton and Pemantle [BP] proved that there is a unique invariant measure p of maximal entropy for this action. This measure has the following property: let T be any tiling of a finite region R, and U, the set of tilings in X extending T (we call UT the cylinder set generated by T). Then p(Ur) only depends on R and not on the choice T of a tiling of R. In this paper we compute h(Ur) explicitly, for any finite tiling T. We will prove in section 5 that: THEOREM 1. There is a function P: Z2 + C with the following property. Let E be any jinite set of disjoint dominos, covering “black” squares bl, . . . , bk and “white” squares wl, . . . , Wk. Then ,Q(UE) is the absolute value of the determinant of the k x k matrix M = (mij), where m;j = P(bi - wj). (H ere b; - wj is the translation vector from square wj to square b;.) We call P the coupling function for ,u. It can be computed explicitly (see Theorem 16 below), and the values of P are in Q $ :Q. A similar result holds for tilings of the plane with lozenges. A lozenge is a rhombus with side 1, smaller angle 7r/3, and vertices in the lattice Z[e 2*i/3] . We define a measure v on the space X’ of lozenge tilings, which is the limit of uniform measures on periodic tilings, and show that its entropy equals the topological entropy of the Z2-action. (For a background on measure-theoretic and topological entropy see [PI.) We conjecture that v is the unique measure of maximal entropy. We compute the coupling function for v in Theorem 9: it takes values in Q $ $Q. We give several applications of these two results. First, it is known that the translation-action of Z2 on dominos or lozenges is topologically mixing (if two events are far enough apart, one can find a single tiling containing both of them). We show here that it is strongly mixing for the measures p and V, that is, events that are far apart are almost independent. In particular we show that the convergence rate to independence is at least quadratic in the distance, that is, for any two finite tilings Tl and T2, and u E Z2,
and similarly for V; see Theorem 11. There is a corresponding lower bound when Tr, Tz are single tiles. Correlations for single tiles have been previously computed by Fisher and Stephenson [FS] and Stephenson [St]. Annales
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Another application is to the computation of the variance of the height function for v. The height function of a lozenge tiling was introduced by Blote and Hilhorst [BH]. We compute for v the variance of the height function in the y-direction (Theorem 15), that is, the variance in height between two vertices separated by distance 71 in the y-direction. This variance is related to the expected number of “contour lines” separating the two points (Proposition 12). The variance in other directions can also be computed with the same methods, although we could not obtain a closed form. On the other hand, domino tilings also have a height function, but we were not able to compute its variance using the methods here. A third application which we will not discuss here is to the computation of entropy ent(s, t) of tilings whose height function has a fixed average slope, that is, the set of tilings of the plane whose height function ht(.. .) satisfies ht(z, y) = ht(0, 0) + sx + ty + o( 1x1, 1~1).This computation is used in [CKP] to give a method of counting the approximate number of tilings of any region, by maximizing an entropy functional over the region. The results of this paper are very general. Using a result of Kasteleyn [K2], we can compute cylinder set measures (for certain measures of maximal entropy) for the space of perfect matchings on any periodic planar graph. Here “periodic” means the graph is finite modulo an action of 2’ by translations. In particular Hurst and Green [HG] showed that the Ising model on a planar graph with nearest-neighbor interactions can be represented as a matching problem on another planar graph (actually they showed this in an equivalent form). Our method therefore allows one to compute the local probabilities for the Ising model. Indeed, methods similar to those in this paper were used in Montroll, Potts and Ward [MWP] to compute long-range correlations for the Ising model. The coupling function which we define is analogous to the Green’s function for a resistor network (see [BP]). Indeed, subdeterminants of the Green’s function matrix give probabilities of finding subsets of edges in a random spanning tree. There is a well-known connection between domino tilings and spanning trees: Burton and Pemantle [BP] have found, via enumeration of spanning trees, the measures of certain cylinder sets for dominos. Our method, which is different, gives the measure of all cylinder sets and is adaptable to planar graphs where the spanning tree connection is not known to hold. The structure of this paper is as follows. Section 2 discusses background: we show how to compute the number of tilings as a determinant, and we define ,u, v and the height function. In section 3 we show how to compute the cylinder set measures for finite planar graphs. In section 4 we compute Vol. 33. Ilo s-1997.
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explicitly the coupling function for v and give the applications. We chose in this section to concentrate on the case of lozenges rather than dominos since lozenges seem to be less prominent in the mathematics literature. Section 5 restates the results of section 4 for dominos.
2. BACKGROUND 2.1. Matchings
on finite planar graphs
Let G be a graph. Two edges of G are disjoint if they have no vertices in common, A perfect matching of G is a set of disjoint edges which contains all the vertices. If G is finite, connected and bipartite, we say G is baIanced if there are the same number of vertices in each half of the bipartition. Let Z2 denote the graph whose vertices are Z2 and whose edges join every pair of vertices of distance I. It is easy to see that domino tilings of the plane correspond bijectively to perfect matchings of Z2. The graph Z2 is bipartite: color the vertex (x,9) black if z + y E 0 mod 2, otherwise color it white. Note that then a domino “covers” exactly one vertex of each color. Similarly, lozenge tiiings correspond bijectively to perfect matchings of a particular bipartite planar graph H, the “honeycomb” grid. The graph H is the l-skeleton of the periodic tiling of the plane with regular hexagons (see Fig. 1). Rather than use lozenges as defined in the introduction, we
Fig. 1. - A matching of the corresponding
on the graph H and some lozenges (dotted lines).
LOCAL STATISTICS OF LATTICE DIMERS
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adopt the following coordinates. Let 2 = $ - 9 and i = $ + q. The set of vertices of H is V = V, U VI, where VO = {a? + b$ ( a, b E a} are the “black” vertices, and VI = 1 + V. are the “white” vertices. There is an edge between every pair of vertices at unit distance. We will denote the vertex a? + bi by the triple (a, b, 0), and a vertex ai + b$ + 1 by the triple (a, 6, 1). Hopefully denoting vertices with ordered triples will not confuse the reader. Note also that a lozenge has side & in these coordinates. Kasteleyn [Kl] and Temperley and Fisher [TF] independently computed the number of perfect matchings of an m x R. grid (in other words, the number of domino tilings of an m x n rectangle). Kasteleyn in [Kl] also computed the number of matchings on a toroidal graph in the case of dominos. Later, in [K2], he showed how to compute the number of perfect matchings of general planar graphs (thereby including the case of lozenges). His method is to count tilings using Pfaffians. We outline here an adaptation of his proof for the case of a simply-connected subgraph of H. We say that a subgraph of H is simply connected if it is the l-skeleton of a simply connected union of “basic hexagons” of H. The following is a special case of the results of [K2]: THEOREM 2. - Let H’ be a Jinite simply connected subgraph of H. The number of pe$ect matchings of H’ equals dm, where A is the adjacency matrix of H’.
Here is a sketch of the proof: since the graph H’ = (V’. 23’) is bipartite, we have V’ = Vd U V,’ (where Vi c V,, Vi c VI) and E’ c Vd x V{. Order the vertices V’ so that those in Vi comes first. We then can write
where B cardinality matchings matrix B
: RlcGl --+ BBI”:l. We can assume that Vd and V{ have the same n (that is, H’ is balanced), for otherwise there are no perfect (and det (A) is clearly 0). Then the determinant of the square is
(2.2)
detB=c sgn(~)a,(l) . .G,~(~~,)
where A = (aij) and the sum is over bijections c vr/;r+ Vi. Each nonzero term in the sum corresponds to a matching. One need only check (and this is the most interesting part, although we won’t do it here) that each term has the same sign. Thus 1det(B)I = dm = # of matchings. Vol. 33, no 5-1‘397.
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This same method also works for dominos, except that one must modify the adjacency matrix to make the signs in (2.2) work out right. One way to modify it is to put weight i = fl on vertical edges [WI. It is slightly more complicated to count matchings on a graph embedded on a torus. Kasteleyn showed that, for a graph on a torus, one can count matchings using a sum of 4 Pfaffians (see Lemma 7 below). 2.2. The measure of maximal entropy An important issue not dealt with by Kasteleyn is the appropriate sense in which random tilings of large finite regions in the plane resemble random tilings of the whole plane. Indeed, as the work of [EKLP] shows, the local configuration of the boundary of a region has a drastic effect on the number of perfect matchings, and hence one must take care when computing the measure p as a limit of measures on tilings of finite regions. A tiling with free boundary conditions, or simply free tiling, of a region R is a tiling of R, where the tiles are allowed to protrude beyond the boundary of R. The topological entropy of domino tilings is by definition
where 5’1, is the set of free tilings of a k x Ic square. Here the square regions Sk can be replaced by any sequence of “sufficiently nice” regions, for example, convex regions containing squares of size tending to infinity. In this case one divides by the area of the k-th region, not by k2. The same definition also applies to lozenge tilings (replace the k x Ic square with for example the set of vertices in a k x Ic square centered at the origin). The entropy H(,u’) of a translation-invariant tilings of the plane can be defined by
measure 11’ on domino
We have H(p’) 5 Htop for any translation-invariant measure $ on X, since, as the reader may readily show, for a finite set of real numbers {ui} satisfying ai > 0 and C ai = 1, the quantity - C ai log ai is maximized when the ai are equal. Andes
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For any measure p’ defined on the finite set X(S) region S, the entropy (per site) of $ is
of tilings of a finite
where /l’(s) is the probability of the tiling s. 2.2.1. The domino measure. - Burton and Pemantle showed: THEOREM 3 ([BP]). - There is a unique measure of maximal entropy p for domino tilings of the plane. The entropy of p equals the topological entropy J&op(X).
Let Zk,, be the quotient of Z2 by the lattice of translations generated by (am, 0) and (0,271). Then Z&,, is a graph (on a torus) with 4mn vertices. Let CL~,~ denote the uniform measure on perfect matchings of Zk,,. Let ,u* be a weak limit of pm,n as m, n -+ 00. We claim that p* = ~1. To see this, Kasteleyn [Kl] showed that the entropy of pL,,, converges as m, n -+ cc to the entropy of CL. So it suffices to show the limit of the entropies is the entropy of the limit. Let Wk by a k-by-k window in Z2 (or on ZL,, when m and n are larger than k). Let pL,,,(Wk) denote the restriction (projection) of pm,n to Wk. Then p,,,(Wk) converges to ,u* (Wk) by weak convergence. Since there are only a finite number of configurations on Wk, lim H(i-hn(Wk)) nL,R’cc
Since ff(~h,~(Wk))
2 H(p.,,,,)
= ff(~*(Wk)).
we have
Taking limits over Ic gives H(p*) = Htop and therefore by the theorem p* = p. We would like to thank Jeff Steif and Jim Propp for the idea behind this argument. 2.2.2. The lozenge measure. - Burton and Pemantle’s proof of Theorem 3 above uses a connection between spanning trees and domino tilings. Lozenge tilings are in fact also in bijection with a set of spanning trees, or rather, directed spanning trees (“,arborescences”) on a directed triangular grid, see [PW]. However it is not evident how to adapt the proof of [BP] to this case. We will show nevertheless that the uniform measures on certain Vol. 33. no S-1997.
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toroidal graphs converge to a measure I/, and their entropy converges to the topological entropy. By the argument following Theorem 3, the entropy of 11 equals the topological entropy. The only fact missing is the uniqueness of Y. be the quotient of H by the lattice of translations generated Let &,, by rni and Qj. Then H,,, is a graph with 2mn vertices. Let II,,,, be the uniform measure on matchings of H,,,,. LEMMA 4. - The entropy of u,,,, converges to the topological lozenge tilings of the plane.
entropy
qf
Proof - The proof is a simple variant of an unpublished trick due to G. Kuperberg. Let F, be the set of free lozenge tilings of the region R,, where R, consists of an equilateral triangle of side nfi (recall that a lozenge has side length a). Then F, is partitioned into equivalence classes, two tilings being equivalent if they have the same set of tiles which protrude beyond the boundary. The number of equivalence classes is less than C” for some constant C, so some equivalence class B, must satisfy
Reflect the region R, along one of its edges c. Given any two tilings 7’1, T2 E B,, there is a tiling of the union of R, and its reflection, where the R, is filled with Tl and its reflection is filled with the reflection of Tz. This is because the tiles which protrude across the boundary edge c are fixed under this reflection. Take the union of 18 reflections of R, so as to make the parallelogram of Figure 2. Given any 18 elements of B,, we can fill them in the 18 triangles
Fig. 2. - The 18 triangles make a torus. Ann&s
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to make a tiling of this whole parallelogram. Furthermore, the configurations of tiles protruding beyond opposite boundaries of this parallelogram are translates of each other. As a consequence we can glue opposite sides of the parallelogram together to make a tiling of the 3n by 3r1 torus (tilings of which correspond to perfect matchings of H3n,3n). We therefore have (where N,,, is the set of tilings of Han,zl,)
This implies that
which completes the proof. q We will see later that H(v,,,) and N(v,,,) have the same limit as m, n + cx), so the restriction to the tori E&, is unimportant. In fact we will show in section 4.4 that there is a unique weak limit to the v,,,!. Let 11 be this limit. We make the following CONJECTURE. - The measure v is the unique measure ,for lozenge tilings of the plane.
of maximal entropy
2.3. Height fmctions For any perfect matching of a simply connected subgraph H’ of H one can associate an integer-valued height fwwtim to the faces of H’. (By face of H’ we mean a basic hexagon which contains an edge of H’.) Let T be a perfect matching of H’. Pick a face zu arbitrarily and assign it height h(zO) = 0. For any other face z, take a path x0, x1, . . . ,x, = x of faces, where for each 3, zj+l is adjacent to ‘cj along an unmatched edge. Let UI = e2xi/3. Then h(~j+~) = h(zj) f 1, where h increases by 1 if ~;j+l - zj is in directions i, iw, or iw2 and h decreases by 1 if zj+l - :~j is in directions -‘i, -iw or -iw2. This defines a height on each face, which can easily be shown to be independent of the choice of path to that face. Figue 3 shows a height function for the matching in Figure 1. For dominos there is a similar definition; see [T]. For a simply connected subgraph H’ which has a perfect matching, the height function along the faces bounding the region (that is, faces outside H’ but containing an edge of H’) is well’defined up to an additive constant and is independent of the matching [T] (rather, this is true if the union of H and these faces is still simply connected). Vul. 33, Ilo 5.1997.
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Fig. 3. - Height
3. THE MEASURE
function
for lozenges.
OF A CYLINDER
SET
The results in this section are stated for lozenges, but they apply verbatim for dominos, on condition that one puts weight i = fl on vertical edges in the adjacency matrix B. Let H’ be a finite, balanced, simply connected subgraph of H, with vertices {q , . . . ,w,} E VF and {vi,.. . , w;} E VI. Let E = {ei,. . . ,ek} be a subset of disjoint edges of H’, with ej = wP,vi3 for j = 1, . . . , k. In the formula (2.2), each nonzero term in the sum corresponds to a perfect matching; those terms whose matchings contain all the edges in E are precisely the terms which contain the subproduct
The sum of the terms containing this product is (up to sign) the product of aE with the determinant of the cofactor BE, that is, the determinant of the matrix obtained from B by removing all rows pl, . . . ,pk and all columns qi,..., qk. This can be proved inductively as follows. Expand det(B) along the row pl: det(B)
= (-l)pl+l
(apI1 det Bpl,l - up12 det Bpl,2 +. . . fapln Ann&s
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The only term that contains aplql is the term for column 41, which is the term (-l)pl+qlapIql det BpIql. Now expand BPIql along row pz, and so on. Thus the sum of the terms containing aE is (-1)x l-‘J+qJaE det( BE). Since laE/ = 1 this proves PROPOSITION 5. -
The number of peeect matchings containing all edges
in E is
Later on the sign of det (BE) will be important to us. We note therefore that the number of perfect matchings containing all the edges in E is (-l)~(p~+y~)a~
det(Bs)sign(det
B).
In a more general setting (e.g. in [CKP]) one might wish to consider weighting the edges non-trivially, in which case the formula (aE det(BE)I counts each matching according to the product of the weights on its edges. Note that the quantity det(BE) does not depend on the actual set of edges in E, but only on the set of vertices involved. This shows, as expected, that each matching of the vertices of E can be extended to the same number of perfect matchings. The matrix BE is an (n - k) x (n - k) cofactor of B; its determinant is equal to det( B) times the determinant of the k x k cofactor of the inverse of B, (B-~)E-, where E” is the set of rows and columns not involved in E. So we have THEOREM 6. -
The number of per$ect matchings containing all edges
in E is 1det((B-l)p)
det(B)I.
That is, I*ff’(UE)
=
1 det((Bvl)E*)I.
More precisely, PHI
= (-l)~pJ+qJaEdet((Bsl)E*).
The advantage of this over Proposition 5 is that the computation is a determinant of size k rather than R. - k. In particular if E consists of a single edge V~IJ~,the probability that that edge is in a random matching is the absolute value of the ij-th entry of B-l. The rest of the paper consists of the computation of B-l and applications of Theorem 6. Vol. 33, no 5-1997.
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4. LOZENGE
TILINGS
4.1. The torus Recall that IS,,,, is the toroidal graph H/(mZd + n@), and the vertex r? + si f t is designated by the triple (T, s, t) E Z/mZ x iI/& x (0, l}. Let Al be the adjacency matrix of H,,,,. Let A2 be the matrix obtained from Al by changing the sign of the entries corresponding to edges from (m - 1, k, 1) to (0, k, 0) f or each k E Z/G!. Let A3 be the matrix obtained from Al by changing the sign of the entries corresponding to edges from (k, 78- 1,l) to (k, 0,O) for each k E Z/ma. Let A4 be the matrix obtained from Al by changing the sign of both these sets of entries. Let Br , &, Bs, BJ be the matrices obtained from Al, AZ, AS, A4, respectively, as in (2.1). Following Kasteleyn, on condition that m, n are both even, we have LEMMA
(4.1)
7. -
Z,,,
The number of perjkt
= f(-
matchings Z,,,
of H,,,
is given by
det B1 + det Bz + det Bs + det Bd).
A term for each perfect matching appears in each of these four determinants. The signs are arranged so that each perfect matching is counted three times with sign + and once with sign - (see [Kl]). For other parities of m and n a similar equation holds but with different signs. Throughout the rest of this section we will assume m and 7~ are both even and neither is divisible by 3. This is simply a matter of convenience; the computations go through in the other cases but require slightly different arguments. Theorem 6 also applies to tilings of the torus. The quantity det B1 det(B,l)E* contains a term for each matching containing ail edges of E. The sign of these terms is the same as the sign of the corresponding terms in det(&) (this easily follows from the proof). This is also true for Bz, Bs, &, and so LEMMA 8. - The number ofpeflect matchings o~T&~ containing a given set of disjoint edges E is given by
(4.2)
f/ - det B1 det(B,t)E.
+ det B2 det(B,l)E*
+ det Ba det(B;l)E*
+ det B4 det(B,r)E*
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4.2. Computation
DIMERS
603
of the eigensystem
The eigenvectors of the Aj are functions f(r, s, t) = C&WY, and f’(r. s, t) = C:Z”W where co = dl + z-1 + W-l Cl&m CL = co c; = -cl (for either choice of sign of the square root), and
The corresponding
zm = 1, ,p zz -1,
wn = 1 wn = 1
for Al for AZ
.P = 1, ,p = -1,
w”=-1 wn=-1
forA forA4.
eigenvalues are
x = COCl X’= -c()q. Consider for example the matrix Al . Let
Ck,P,l
=
1 + 2$ + UI:
c:,,,o = clc,e.o I (any choice of sign for the square roots will do). Then fk,e(r, s, t) = Ck,e,&w: and &(T, s,t) = c~,~,~z~w:S are eigenfunctions. The functions fk,e + fL,e are zero on VI, and run through a basis for all Vol. 33, no s-1997
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R. KENYON
functions on V, (the exponentials) as (k:, !) runs through (0, . . , m - l} x (0,. . . , n. - l}. Similarly the functions fi~,~ - fL,[ are zero on V. and run through a basis for functions on VI. Thus the collection of fk,e and fL,y is a basis of eigenvectors for the matrices Al. In particular we have (4.3) det(Al)
=
=
n
l-j
k=O
e=o
m-l
n-l
k=O
e=o
m-l n
n-l n
k=O
e=o
Ak,eA&e
(1 +
,$W+n
+
e27ee/n)2
(we removed the initial minus sign in this last equality since there are an even number of terms, and also in the second product we replaced k, /2 with -k, 4). Similarly one can compute (4.4) det(A2)
= n(l
+ ,pw+l)h
+ e2wn)(l
+ e-4wl)/m
+
e-27wn)
k,e =
l3 Ic.e
l+e
?ri(2k+l)/m
+
e2rie/n
2 >
and furthermore (4.5) (4.6)
det(A3) = n(l k,e det(A4) = n(l k.e
+ $niklm
+ pW+l)ln)2
+ exi(2k+l)/m + e42e+l)/n)2,
These four determinants are non-zero by our assumption that m and n are nonzero modulo 3 (three complex numbers of modulus 1 sum to zero only if they are a rotation of the set { 1, e2ri/3, e4?r’/3}). 4.3. Computation
of the inverses of I&, . . . , B4
It suffices to invert the A?. Define the vector fio,o,o(?Y7t)
1 if(z y t)=(O,O,O) = { o othe;w;se Ann&s
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Consider first the matrix Al. We can write
~o,o,o(& Y>t> = &
1
fk,e(? Y(/, t> + G,eh Y>t) ck,e,o
k.t
Thus the vector
has the property that
&Po,o,o(TY, t>
= SO,O,O(G YV,t).
The preimages of delta functions at other vertices can be similarly defined: define Pr+,o (z,y,t) = PO,O,O(X - r,y - s,t); then
Thus the function PT,S,o gives the entries of the (r. s, 0)-column of the matrix AlI. For notational simplicity we will write P(l)(z, y! t) = Po,o,o(z, y, t) (the superscript (1) refers to the fact that it comes from matrix A,; note that I’(‘) also depends on m and n). If we simplify (4.7) above: .fk,&,
y,t) - G:,P(~>Y>t) >
xk.P
,
we see that P(l) (z, y, 0) = 0 for all 2, Y. On the other hand plugging in for f, c, A, zl, wr we have (4% Similar expressions hold for the inverses of AZ: As, Aa, except that the sum is over a different set of angles. For example for A2 we have eKi(2k+1)~JlrLe2~iVy/n
P(2)(z, y, 1) = +& ck Vol. 33, n” 5.1997
e 1 +
e--rri(2k+l)/rn
+
(i-2rie/7L.
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4.4. The limiting
measure
Here we compute the limit of the ratio of (4.2) and (4.1), that is, the limit of the cylinder set measures v,,,( UT). We still assume m, n $ 0 mod 3 so that the determinants in (4.3)-(4.6) are nonzero. We will show below that the quantities det((BJT1),!+) forj = 1,2,3,4 all converge to the same value as m, 71 -+ co. Then (if we remove the absolute value sign in (4.2)) the ratio of (4.2) and (4.1) is a weighted average of these four quantities, with weights f det Bj/Z,,,,. These weights are all in the interval [-1, l] since Z,,, 2 (det By/ for each j: recall that Z,,,, and 1det Bj ( count the same objects except that some signs in 1det Bj/ are negative. Since the weights sum to 1, the weighted average is also converges to the same value as each det( ( BJT1)p ). It suffices to show that, for fixed X, y, for each j = 1,2,3,4 the quantities P(~)(x, g/,1) tend to the same value (recall that the matrix BJ” has entries which are the values of P(j)). For a fixed integers 5, y, the function
is continuous on ([O, 2n] x [0,27r]) \ { (27r/3,47r/3) U (47r/3,2~/3)}, and has a pole at the two points (2~/3,47r/3) and (47r/3,2n/3). The expression (4.8) and the corresponding expressions for the other P(j) are each Riemann sums for the integral of p over [0,2~] x [0,27r]. Fix 6 > 0 and let N6 be a 6 x a-neighborhood of the pole (&, &) = (2~/3,4x/3). Outside N6 (and the corresponding neighborhood of the other poie) the Riemann sums converge to the appropriate integral. We must show that on NS the sums are small (in the sense that they tend to zero with S as m, n --f co,). On Ns, we use the Taylor expansion of the denominator:
(@,d)ENh ILC mn
ZZ-
N6 1 - ieuieo(B
1
1
mn cN6 Ie-2pi/3(8
=-
- t90) -
- 00) + e-4?ri/3(4
- tie) + O((0
1 mn
cNg
(,-21ri/3(8
_
0,)
:
e-47ri/3(t#,
Annales de l’lnstitut
_
40)l
- 60)2, (t#~ - c$~)~)]
1 +
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‘(’
-
go> 4 -
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(The reason we can take the big-0 term out of the denominator is that (,-2Ti/3(0
-Oo)+
e-4d3
(4 - 4011 2 const(lo
- 001 + 14 - 401)
for 10 - 601 and 14 - q50 ) small; we can then move it out of the summation altogether because of the factor &.) Since m, n f 0 mod 3, we have, in the sum for each P(j), that 8 - Ho is of the form 8 - B. = r( 2 - 3) = 7r& and similarly q5- & = 5, where 5’ and e’ are not multiples of 3. Thus the sums are each bounded by one of the form -1
mn c
where the sum is over
This sum is easily shown to be bounded independently of m, n and tending to zero as 6 ---t 0: this follows from the integrability near the origin of the functh ,r+eAi3y, with respect to f&&y. THEOREM 9. - For ajnite matching E covering black vertices { bl , . . . . bk } and white vertices { wl? . . . , w~c}, we have v(UE) = 1[let, P(w; - bj)], where 2n P(x,
2m
y; 1) = -& .o Is
0
ei"8eiy&dedq 1 + e-i8 +
e-im'
Pro06 - This follows from Theorem 6 and the convergence argument above. 0 4.5. The coupling function
for lozenge tilings of the plane
We call P(x, y, t) the coupling function for lozenge tilings of the plane. Let .z = ePie and w = ePid. Then we obtain the contour integral
The function P has all the symmetries of the graph H, that is, (4.10)
P(x,y,l) = P(y,x,l)
Vol. 33. Ilo s-1997.
= P(-z
- y - l,z, 1) = P(y,-3:
- 9 - 1,l)
= P(-,
- y - l>?/,l)
- y - 1,l).
= P(:c,--3:
608
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These are obtained by noticing that (4.9) is invariant under interchanging z and w, or under the substitution (z,ru) -+ (w-l, ZW-l), or under combinations of these. We can explicitly evaluate the integral (4.9) as follows. This is slightly easier to calculate in the case z _< - 1. The other values can then be obtained by (4.10). If we fix w and integrate over Z, there is a pole inside the unit circle when 11 + WI < 1, that is, when -q5 E (2~/3,4~/3). The residue of z-“-r/(1 + z + w) at this pole is (-1 - w)+-l. For -q5 6 (2~/3,4~/3) there is no pole inside the circle and therefore the integral is zero. So when x 5 -1, e4*t,3
(4.11)
P(x, y, 1) = 2(-l)-“-’
s &2TZ/3
W
-y-y1
+ w)-“%w.
For example when x = -1 we have (pyp
(4.12)
_
e4TTiy/3
Y
>
- 4 27ry ’
where cy = 0, 1, -1 if y E 0, -1,1 mod 3. Some values of P(x, y, 1) are plotted in Figure 4. To compute these values, start from (4.12), which, along with the symmetries (4.10) gives the value on 3 lines, and use the
Fig. 4. - Values
of P(z, Andes
y, t). Here de l’lnstitut
T = &/(h).
Henri
Poincart!
- Probabilitt3
et Statistiques
LOCAL
STATISTICS
OF LATIICE
609
DIMERS
fact that the sum of the three values adjacent to any vertex (except the origin) is zero (since AP = 0 at these vertices). For example the equation P(-k,O?
1) + P(-k
determines P(-k, 0,l) before that P(-k,-1;l)
+ 1, -1: 1) + P(-k
+ 1.0.1) = 0
for k > 0 inductively for increasing k (note as = P(-l,-k,l) = $).
4.6. Some local probabilities
for lozenges
From Theorem 6 and equation (4.11) with z = - 1, y = 0, the probability of the edge (0, 0,O) (0, 0,l) being in a random matching is given by P(0, 0,l) = $, as expected by symmetry. The probability of both the edges (O,O, O)(O, 0,l) and (0, l,O)(O, 1,l) occurring is given by
Po,o,o(O, 0, 1) ~0,0,0(07 171) PO.l,O(O,
0, 1)
The probability
~0,1,0(07
171)
z 13%. =
of the three edges
CO,% O)(O,(Al), ((Al, ‘4-L 1;1) and(-l, l, 0)(-l, ($1) (which bound a hexagon) all occurring, is:
(Such a hexagon corresponds to a local minimum for the height function.) Note from (4.12) that P(-3k,3k, 1) = P(-1, -3k, 1) = 0. As a consequence, the set of edges E = {en}nEZ where e, is the edge (-3n, 3n, O)(-3n, 3n, 1) is an “independent” set: the probability of any fixed subset of size k occurring in a random matching is exactly ( 1/3)L. Is there any simple explanation for this fact? We can also compute long-range correlations. The probability of the horizontal edge (0, 0, O)(O, 0,l) and the horizontal edge (-71,n, 0)(-n, n, 1) both occurring is given by the determinant 1 5
P(n, Vol. 33, no 5.1997.
-n,
P(
1)
-n,
n,
1 3
1) =
f
-q-71,72,1)”
610
R. KENYON
By (4.10) and (4.12), P(- ?L,71, 1) = P(-1, -n, 1) = I,&/. the probability of these two edges both occurring tends quadratically not faster) to i. More generally we have PROPOSITION
so
(but
10. P(x, y, 1) = 0
( I4 : IYI > . Proof. - We show for z 5 -1 that P(z, y, t) = 0( 6); by the dihedral 6-fold symmetry of P the result will then follow. From (4.11) we have (1+ w(-+-ildW/ =-
2 2r
?r
(2 cos(e/2))-z-‘de
.I &r/3
0 where we used the substitution t = 2cos(0/2). Let Ei and Ea be two finite collections of edges. The probability edges El U Ez are present in a matching is given by a determinant
that
(4.13)
where det(C1) = ~(UE~) and det(C,) = p(U,,). The entries of D1 and D2 are the values of P connecting points of El to Ea. If the distance between El and E2 is 1 n then these entries are all O(i). Let c +J)~l,(l) . a. uka(k) be the expansion of the determinant (4.13). Suppose aie(~) is an entry from the submatrix Di, that is, i 5 IElI and c(i) > J&I. Then there must exist an index j with j > [El/ and c(j) 5 JEil (since c is a bijection), that is, ajOcj) is an entry of Da. We may similarly conclude that each term in the sum for the determinant has the same number of entries from Di as from Dz. Therefore the determinant (4.13) is equal to det(Ci) det(C2) + 0( 3). This proves THEOREM
11. - If the distance between El and E2 is at least n, then
Annales
de [‘Ins&d
Henri
Poincart4
- Probabilit&
et Statistiques
LOCAL
STATISTICS
OF LA’I-IYCE
61 I
DIMERS
Note that the comments before Proposition (10) give a corresponding quadratic lower bound if El and E:! are single horizontal tiles separated by -(3n + l)? + (3n + l)$ An argument similar to Theorem 11 shows that v is mixing of aII orders, that is, given k events El,. . . , & each of which is separated from the others, the joint probability converges to the product of the individual probabilities as the distance between them tends to infinity. 4.7. The height distribution Let h, be the random variable which gives the height difference between the face at the origin (by which we mean, the face above and right of the point (0, 0,O)) and the face at the point C--r&,n, 0) in a random lozenge tiling of the plane. In this section we compute the variance of h,. Since the height is only defined up to an additive constant, we may as well assume that the height on the face at the origin is zero. One motivation for computing this variance is that it is related to the number of cycles in the union of two random matchings as follows. Given two random matchings Mi and Mz, their union is a set of disjoint cycles. The difference of their height functions h (l) - hc2) has the property that it is constant on the connected components of the complement of these cycles, and changes by 3 when crossing any of these cycles. The height difference across a cycle increases or decreases by 3 depending on which of the two possible matchings of the cycle occurs in M1 (with the complementary matching necessarily occurring in Mz). Since the two matchings were chosen randomly, across each cycle the difference of heights increases or decreases by 3 independently of what happens at the other cycles. (Among all pairs of matchings whose union has this particular cycle structure, each matching is equally likely to have either “half’ of each cycle.) Thus for a face f separated from the face at the origin by Ic cycles, the variance in the height difference (that is, the variance g2(Ah(f)) = a2(h,(1)(f)-h(2)(f))) is th e same as the variance of a sum of k independent fair coin flips of values f3. This variance is 9k. Since n/l, and IV2 were chosen randomly, the number y of cycles separating f from the origin is also a random variable. We have a2(Ahly) = 9y, and
“2(Ahly)= WW21v>- Whld2 = E(W)“ld since E(Ahjy) a2(Ah)
= 0. Furthermore,
= E((AIL)~)
= c
P(y)E((Ah)21y) Y
Vol. 33, no 5-1997
= c
P(y)9y Y
= 9E(y).
612
R. KENYON
Thus in general the variance of the height differences at a fixed face (for two matchings) is 9 times the expected number of cycles separating that face from the origin. Since the two matchings MI, it& are chosen independently, the variance of their height difference at f is exactly twice the variance of the height of a single matching at f (note that the height at f has mean value zero, which follows from the fact that the probability of any edge being present is l/3). We conclude 12. - In the union of two random matchings, the expected number of cycles separating two faces fi, f2 is 2s/9, where s is the variance in the height (between fi and fi) of a single random matching. Let E, denote the set of horizontal edges (-k, k, 0)(--k, k, 1) for k = l,...,n. PROPOSITION
LEMMA
13. - The value of h, is equal to n - 3r,, where r, is the number edges from E; in the matching.
of horizontal
Proof. - This follows from the definition of the height function. On the vertical path of faces from (0, 0,O) to (-n, n, 0), the height increases by 1 for edge of E, not present in the matching, and decreases by 2 for every edge present in the matching. So the height difference is (n - T,) - 2r,. 0 Let A4 = M, be the n x n matrix A4 = (mk,j) where
Then as we saw, the determinant of A4 is the probability that all the edges from E, are present in a random matching. By symmetry (4.10), when e > 0,
zq-e,e, 1) = q-1, -e, 1) = =. For example when n = 5 we have (with r = $)
For each subset S c I$, of cardinality k, let MS be the submatrix of M whose rows and columns are those of M indexed by 5’. Then
LOCAL
STATISTICS
OF LATTICE
613
DIMERS
C-G
p3+qj det(Ms) is the probability that the edges corresponding to S are present in a random tiling, where the sum is over S. Since pj = 4.j in our ordering, the probability in question is exactly det (MS). Let p(z) = i? - cQ.P--l + Q2Zn-2 - . . . + (-l)?lQI, be the characteristic polynomial of M. Then CXI,is the sum over all subsets S c E, of size k of det MS, that is, the sum over S of the probability that S is in the random matkhing. that there are exactly Let d2) = cizo Pk zk where ,& is the probability k edges from E, present. A formula of Ch. Jordan [C] relates the (kk t0 the p,: @k=ak-
(‘;‘)ak+l+
(‘;1)(2k+z-....
From this we can easily derive the following: THEOREM
14. -
We have q(z) = (1-
z)“p
&
. q
(
>
Now if p(z) = n~=“=,(z - Xi), then q(z) = fi(l
- x; + X+,z).
i=l
The expected value of T, is c k/& = q’( 1). we find
so q’( 1) = q( 1) c
Xi = c
X; = trace(M)
= n/3.
This is as expected since the probability of each edge (-k, k, 0)(-k, being present is l/3. Similarly, the variance of T, is given by 2 C2(&)
=
xk2&
-
(ck/&)2
=
q'(1)
+
q"(1)
-
f 0
Vol. 33. no 5-1997.
.
k, 1)
614 We have
R. KENYON
d’(4=d4ci#Li (1 - A; + X$)(1A;$- xj + X,2)’
and so q”(1) = C
XiXj = trace(M)2
- trace(M2)
= (n/3)2 - trace(M2).
i#j To compute trace(M2), 1 < i < n is:
note that M is symmetric and the ith row for
The inner product of this row with itself is
. Similar sums (with y replaced with l/3) hold for the rows i = 1 and i = n. Summing these up for i = 1, . . . , n yields (4.14)
trace(M2)
= % + r2 2(n - 1) + 29
+2(n-4 ___
+
2-I -+qn-
42
I)2
Note that
Ann&s
de l’hsfitur
Hemi
Poincarti
Probabilit~s
et Statistiques
LOCAL STATISTICS OF LATTICE DIMERS
615
and similarly
l+;+;+...+rI4 =(I+;+;+...+;>-; (1+;+;+...+IA > = log(r) + O(1) - $log(r/3)
+ 00))
= i log(r) + O(1). Plugging this into (4.14) gives trace(M2)
hd4
= t - 7
+ O(l),
which yields
02(r,) = y
+ O(1).
Finally we find THEOREM 15. - Let h, be the height diference between the origin and the point (-n, n, 0). The expected value of h, is 0. The variance in h, is given by a2(h,) = v + O(f). The only previously known result in this vein is in {CEP], where Cohn, Elkies, and Propp found a 0(&k) upper bound for the height variance on a bounded region.
Proof of Theorem 1. - The proof follows from Theorem 6, the fact that ,h,n --+ JL {see the comments after Theorem 3), and a convergence argument analogous to that preceding Theorem 9. 0 In fact a computation analogous to the case of lozenges yields: THEOREM 16. - The coupling function for dominos is 2rr IS0 vol. 33. no 5-1997.
2x 0
e+@+Y'd')&jd+
2 cos(0) + 2i cost
616
R. KENYON
This statement depends on our choice of weighting the vertical edges of Z2 with weight i. As for lozenges, I’(,, y) is zero for black vertices (i.e. when x + y E 0 mod2). Some values are shown in Figure 5 (where the origin is in the lower left comer).
-it 13 w-w 10 4 A )
i 1 1 ; i.4 4 “(4 -..-...-..;...---..------;------------.-. -;) :O . . . . . . . . . . . . . :0 . . . . . . . . . . . . ..-.... :-I( 2+ rr ) 0
Fig. 5. - Values of P(z, y) for dominos.
One can explicitly evaluate, for 2 2 1, P(2z+1,2s)=(-1)”
;-;
l-;+;-...+(-qZ+r--&
[
)
(
>I
and P(2x, 22 - 1) = -iP(22
+ 1,2X).
From these values (and the diagonal symmetry) the others can be obtained inductively using AP = 0 except at the origin. Ann&s
de I’lnstirut
Henri Poincarc!
- ProbabilitCs
et Statistiques
LOCAL
STATISTICS
OF LA’ITICE
As an example, the probability (0, l)(l, 1) both occurring is
617
of the two edges (0, 0)( 1,O) and
I1 =i I ”4.4
DIMERS
4f
=-,
81
The probability of the two edges (O,O)(l,O) occurring is given by
and (0, l)(O, 2) both
This probability was also computed in [BP], although their stated value is incorrect (they have acknowledged the mistake in their arithmetic). Proposition 10 and Theorem 11 also apply to dominos. However the computation of the variance in the height function cannot be done within this framework since there is no simple analogue of Lemma 13.
ACKNOWLEDGEMENT
I would like to thank Oded Schramm, Jeff Steif, Jim Propp and the referee for many helpful suggestions regarding this paper.
REFERENCES IW
H.
[BP1
R.
Kl WPI
L. H.
uw WW
H. N.
IFSI
M.
WGI
C.
[Kll
P.
IK21
P.
W. J. BL~TE and H. J. HILHORST. Roughening transitions and the zerotemperature triangular Ising antiferromagnet, 1. Phys. A Vol. 15, 1982, L631-L637. BURTON and R. PEMANTLE, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer impedances, Ann. Probnb., Vol. 21, 1993, pp. 1329-1371. QMTET, Advanced combinatorics, D. Reidel Publishing Company, Dordrecht. COHN and N. ELKIES, J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J., Vol. 85, 1996, pp. 117-166. COHN and R. KENYON, J. Propp, in preparation. ELKIES, G. KUPER~ERG and M. LARSEN, J. F’ropp,Alternating sign matrices and domino tilings. J. Alg. Combinarorics, Vol. 1, i992, pp. 1 Ii-132. E. FISHER and J. STEPHENSON, Statistical mechanics of dimers on a olane iattice II. Dimer correlations and monomers, f/z~s. Rev., Vol. 132, 1963, ip. 141 I-1431. A. HURST and H. S. GREEN, New solution of the Ising problem for a rectangular lattice, J. Chem. Phys., Vol. 33, 1960, pp. 1059-1062. W. KASTELEYN, The statistics of dimers on a lattice, Physica, Vol. 27, 1961, pp. 1209-1225. W. KASTELEYN , Graph theory and crystal physics, in Graph Theory and Theoretical Physics, Academic Press, London 1967.
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R. KENYON E. W. MONTROLL, R. B. Ports and J. C. WARD, Correlations and spontaneous magnetization of the two-dimensional Ising model, J. Math. Phys., Vol. 4, 1963, p. 308-322. K. PETERSEN, Ergodic theory, Cambridge studes in advanced mathematics 2, Cambridge university press, Cambridge, 1983. J. PROPP and D. WILSON, in preparation. J. STEPHENSON, king-model spin correlations on the triangular lattice, J. Math. Phys., Vol. 5, 1964, pp. 1009-1024. W. THURSTON , Conway’s tiling groups, Am. Math. Monthly, Vol. 97, 1990, pp. 757-773. H. TEMPLERLEY and M. FISHER, The dimer problem in statistical mechanics - an exact result. Phil Mug., Vol. 6, 1961, pp. 1061-1063. T. Wu, Dimers on rectangular lattices, J. Math. Phys, Vol. 3, 1962, pp. 1265-1266. (Manuscript
received April Revised FeQruaq
22, 1996; 3, 1997.)