Domino tilings of cylinders: connected components under flips and normal distribution of the twist

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a 2x2x1 rectangular cuboid. We are particularly interested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixed quadriculated disk. In dimension 3, the twist associates to each tiling $t$ an integer $Tw(t)$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $D$ is regular if, whenever two tilings $t_0$ and $t_1$ of $R_N$ satisfy $Tw(t_0) = Tw(t_1)$, $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. Many large disks are regular, including rectangles $D = [0,L] \times [0,M]$ with $LM$ even and $L,M \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $R_N = D \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $T_0$ and $T_1$ of $D \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to 1 if and only if $D$ is regular. Under a suitable equivalence relation, the set of tilings has a group structure, the domino group. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $R_N = D \times [0,N]$, particularly for large $N$.


Introduction
A quadriculated region D ⊂ R 2 is a planar quadriculated disk if D is the union of finitely many closed unit squares with vertices in Z 2 and D is homeomorphic to the closed unit disk. We always assume that our quadriculated regions D are balanced (equal number of white and black unit cubes). A quadriculated disk D is nontrivial if it has at least 6 unit squares and at least one square has at least three neighbours. A domino is the union of two adjacent unit squares. A (domino) tiling of D is a set of dominoes with disjoint interiors whose union is D; the set of domino tilings of D is denoted by T (D). From a graph-theoretical perpective, we identify D with a simple graph whose vertices are unit squares (or, alternatively, their centers) and whose edges are dominoes (or, alternatively, unit segments joining the centers of two adjacent squares). In this language, a tiling is a perfect matching.
Similarly, a cubiculated region is a set R ⊂ R 3 which is a union of finitely many unit cubes with vertices in Z 3 . A (3D) domino is the union of two adjacent unit cubes and a tiling (of R) is a set of dominoes with disjoint interiors whose union is R. A cylinder is a cubiculated region of the form R N = D × [0, N ] where D is a quadriculated disk. Let T (R) be the set of domino tilings of R. Again, R can be identified with a bipartite graph, dominoes with edges and tilings with perfect matchings. We follow [3,4,7] in drawing tilings of cubiculated regions by floors, as in Figure 1. Vertical dominoes (i.e., dominoes in the z direction) appear as two squares; we leave the right square unfilled.
A flip is a local move in either T (D) or T (R) from one tiling to another: remove two adjacent and parallel dominoes and place them back in a different position. An example of a flip is shown in Figure 1. If D ⊂ R 2 is a quadriculated disk then T (D) is connected under flips ( [8], [6]). For t 0 , t 1 ∈ T (R), we write t 0 ≈ t 1 if t 0 and t 1 are in the same connected component under flips, i.e., if they can be joined by a finite sequence of flips. One of our aims is to study the connected components of T (R) under flips. For a fixed balanced quadriculated disk D, let R N be the cylinder D × [0, N ]. Two tilings t 0 ∈ T (R N 0 ) and t 1 ∈ T (R N 1 ) can be concatenated to define a tiling t 0 * t 1 ∈ T (R N 0 +N 1 ). For even N even, there exists a tiling t vert,N ∈ T (R N ) with all dominoes vertical. Assume N 0 ≡ N 1 (mod 2) and t i ∈ T (R N i ); we write t 0 ∼ t 1 if and only if there exists N ≡ N 0 (mod 2), N ≥ max{N 0 , N 1 } such that we have t 0 * t vert,N −N 0 ≈ t 1 * t vert,N −N 1 . Given a balanced quadriculated disk D, we define the (full) domino group G D : its elements are ∼-equivalence classes of tilings and its operation is * , the concatenation. In [7,2] we construct a finite 2-complex C D for which G D = π 1 (C D ) is the fundamental group, and therefore finitely presented. We review the construction of the domino group in Section 2.
Given a tiling t of R N , we define in [1,3,4,7] the twist Tw(t) ∈ Z. One of the most fundamental properties of the twist is that it is preserved by flips, so that t 0 ≈ t 1 implies Tw(t 0 ) = Tw(t 1 ). There are other local moves (such as the trit) which change the twist in a predictable way. We have Tw(t vert ) = 0 and Tw(t 0 * t 1 ) = Tw(t 0 ) + Tw(t 1 ). The twist is therefore a group homomorphism Tw : G D → Z. This homomorphism is surjective for nontrivial D.
A quadriculated disk D is regular if and only if for any tilings t 0 , t 1 of R N Tw(t 0 ) = Tw(t 1 ) implies t 0 ∼ t 1 (but not neccessarily t 0 ≈ t 1 ). It is not hard to check that D is regular if and only if G D ≈ Z ⊕ (Z/(2)) (the first coordinate is the twist, the second one is the parity of N ). A vague conjecture is that "large" quadriculated disks are regular. In Theorems 1 and 2 we restate the main results of [7] with some extra information (which is also proved in that paper). Here, F 2 is the free group in 2 generators a and b and F 2 Z/(2) is the semidirect product defined by the involution a → b −1 → a. Theorem 2. [7] Let D be a regular quadriculated disk containing a 2×3 rectangle. Then there exists M (depending on D only) such that for all N ∈ N and for all Given R, a random tiling of R is a uniformly distributed random variable T : Ω → T (R). The following conjecture is formulated for boxes but should hold in greater generality. There is some empirical evidence towards it (but it is hard to tell how significant it is; see Figure 2 for a sample).  converges in distribution to a normal distribution centered at 0.
2. Two tilings with the same twist can almost always be joined by a finite sequence of flips: In this paper we prove related results for tall cylinders. Thus, L and M are kept fixed while N goes to infinity. Also, the theorem is restricted to cylinders (not just boxes). In this introduction we state simpler results, with an emphasys on the main examples: regular disks and the twist. In Section 2, after reviewing the contents of [7], we state more general results. The proofs are not significantly different. Our first main result establishes normal distribution of the twist for tilings of nontrivial cylinders.
This roughly corresponds to a special case of the first item of Conjecture 1.1. The normal distribution is illustrated in Figure 2; there is a similar figure in [7]. Convergence in distribution follows from the statement, but is easier and proved first in Theorem 8, stated in Section 2 and proved in Section 5. Theorem 3 is a special case of Theorem 9, also to be stated in Section 2 and proved in Section 6. The next result assumes D ⊂ R 2 to be regular. We then give a description of ≈-connected components of T (R N ). We are particularly interested in the larger components. Let M be as in Theorem 2. A component C ⊆ T (R N ) is fat if it includes at least one tiling with at least M vertical floors; a component in thin otherwise. For a component C with t ∈ C, we write Tw(C) = Tw(t).
Theorem 4. Let D ⊂ R 2 be a regular quadriculated disk. There exists constants c D ∈ Q, b ∈ R andc ∈ (0, 1) such that, for all N ∈ N, the properties below hold.
3. For t ∈ Z, there exists at most one fat component F N,t with Tw(F N,t ) = t.

4.
Let θ N be the number of tilings in all thin components of T (R N ). Then, as N goes to infinity, The constant c D is the same one introduced in Section 11 of [7]; see particularly Lemma 11.1 (restated below as Lemma 7.1). There are many examples of quadriculated disks D ⊂ R 2 for which T (R N ) can be shown to have many thin connected components, even many connected components with a single tiling. The examples in Figure 3 below and in Figure 7 in [7] should make this clear. It would be interesting to have more precise results, perhaps including estimates for the sizes of thin connected components. Our next result has been announced (bot not proved) in [7]. It follows easily from the other theorems in this paper. Consider a nontrivial quadriculated disk D ⊂ R 2 . There is an obvious homomorphism G D → Z/(2) taking t ∈ T (R N ) to N mod 2: let G + D < G D be its kernel. Consider the restriction Tw : G + D → Z and its kernel ker(Tw) < G + D . We have | ker(Tw)| = 1 if and only if D is regular. If ker(Tw) is infinite, 1/| ker(Tw)| = 0. The simplest local move after the flip is the trit. We consider three dominoes in three different directions whose union is a 2 × 2 × 2 cube minus two unit cubes in opposite corners. Remove the three dominoes and place them back in the only other possible way. Two examples of trits are shown in Figure 4. It is shown in [1,4] that if two tilings t 0 and t 1 are joined by a trit then Tw(t 1 ) = Tw(t 0 ) ± 1; the sign depends on the orientation of the trit. The following result shows that almost any two tilings can be connected by flips and trits. In most cases, if t a , t b ∈ T (R N ) there exists a path of flips and trits from t a to t b with precisely | Tw(t a ) − Tw(t b )| trits. Theorem 6. Let D ⊂ R 2 be a regular quadriculated disk. Assume furthermore that D contains a 2 × 3 rectangle. Let c ∈ Q, b ∈ R be as in Theorem 4. Let F N,t be the fat components under flips of T (R N ). Then there existsb ∈ R,b ≥ b, with the following property.
If t 0 ∈ Z ∩ [−cN +b, cN −b − 1] then there exist tilings t 0 ∈ F N,t 0 and t 1 ∈ F N,t 0 +1 such that t 0 and t 1 are joined by a trit.
The set T (R N ) has a giant component under flips and trits G ⊆ T (R N ). More precisely, there existsc ∈ (0, 1) such that |T (R N ) G| = |T (R N )| o(c N ).
In Section 2 we review some more results, particularly from [7], and state the other main results of this paper, Theorems 7, 8 and 9. These results are more general and imply Theorem 3. Their statements make essential use of the domino group G D . Section 3 contains the first results concerning random tilings. This is closely related to studying random paths in a finite graph and uses the Perron-Frobenius Theorem. In Section 4 we use the language of homology to obtain algebraic expressions for, say, the number of tilings t of R N with Tw(t) = t (in terms of N and t). Sections 5 and 6 are the most technical and contain the proofs of Theorems 8 and 9. Notice that the proof of Theorem 9 uses Theorem 8, but not its proof. Section 7 contains the proofs of Theorems 4, 5 and 6: by this point, the proofs are short and simple. Finally, Section 8 list a few remarks and related open questions.
The author thanks Juliana Freire, Simon Griffiths, Caroline Klivans, Pedro Milet and Breno Pereira for helpful conversations, comments and suggestions. The author is also thankful for the generous support of CNPq, CAPES and FAPERJ (Brazil).

Review and results
In this section we review notation and results, particularly from [2,7]. We also state the other main results of this paper, which are harder to state than Theorems 3 and 4. Our focus in the introduction, as in [7], was in regions of dimension 3. As in [2], the general construction below is also valid for regions of dimension n ≥ 4. We then assume D ⊂ R n−1 to be a balanced, contractible, bounded cubiculated region and write R N = D × [0, N ]. Remember however that for tilings t of R ⊂ R n , n ≥ 4, the twist Tw(t) is an element of Z/(2) (not of Z).
A plug is a subset p ⊆ D which is the union of an equal number of black and white unit squares (or cubes); a plug is sometimes confused with a set of unit squares (or cubes). If D has 2k = |D| unit squares, the set P of all plugs has 2k k elements. The empty set is the empty plug p • ∈ P and D is the full plug p • ∈ P. We say two plugs p 0 and p 1 are disjoint if their interiors are disjoint. Given two disjoint plugs p 0 and p 1 , D p 0 ,p 1 ⊆ D is the union of all unit squares contained in D with interior disjoint from both p 0 and p 1 . Thus, D p 0 ,p 1 is a balanced quadriculated region, possibly disconnected and not necessarily tileable. A (full) floor is a triple f = (p 0 , f * , p 1 ) where p 0 and p 1 are disjoint plugs and f * is a tiling of D p 0 ,p 1 ; a floor is vertical if f * = ∅ (see Figure 5). Informally, a floor is what we draw for each floor of a tiling of R N = D × [0, N ], as in Figure 1. A cork is a region of the form a tiling t of R N ;p 0 ,p N is identified with a sequence of plugs and floors: t = For a tiling t of R N , let vert(t) be the number of vertical floors of t. The following result is a corollary of Theorem 2 and is essentially the same as Corollary 12.1 in [7]. A similar result holds for D ⊂ R n−1 , n ≥ 4.
The set of vertices of the complex C D is P. Each floor f = (p 0 , f * , p 1 ) is an undirected edge joining p 0 and p 1 . Notice that tilings of D are loops from For sufficiently large N , all entries of A N are strictly positive.
Flips define the 2-cells of C D : horizontal flips are bigons and vertical flips are squares. If t 0 ∈ T (R N 0 ;p 0 ,p 1 ) and t 1 ∈ T (R N 1 ;p 0 ,p 1 ) are interpreted as paths in C D , we have t 0 ∼ t 1 if and only if the two paths are homotopic with fixed endpoints. The domino group G D = π 1 (C D ; p • ) is the set of tilings of R N (for all N ) modulo the equivalence relation ∼. Vertical tilings represent the identity element in G D . The group operation is * , the concatenation of tilings (or paths). The structure of G D is very informative.
We are ready to state another of our main results.
The proof of Theorem 7 is given in Section 3.
For a sequence of random variables such as (X N ), assuming values in a fixed finite set, the notion of convergence is easy. Indeed, we are merely stating that, for any x ∈ G D /K, we have We shall soon discuss the more complicated case of random variables assuming values in infinite sets.
Notice that we do not assume that K is a normal subgroup; the quotient space G D /K in the statement above can be taken to be either the left coset space {gK; g ∈ G D } or the right coset space {Kg; g ∈ G D } (both have n elements). The following corollary illustrates some uses of Theorem 7.
Corollary 2.2. Consider a nontrivial quadriculated disk D ⊂ R 2 and a random tiling T N of R N . Consider a fixed positive integer n ∈ N * . Then, for any a ∈ Z, Proof. For n ∈ N * , let ψ : G D → Z/(n) be defined by ψ(t) = Tw(t) mod n. Take K = ker(ψ). The first limit in the display follows directly from Theorem 7. For the second limit, apply the first one with n = 2 k and let k go to infinity.
For the next theorems, random variables assume real values. The first result uses the concept of convergence in distribution; we shall further comment on this concept both in and after the statement.
Theorem 8. Let D be a nontrivial balanced quadriculated disk. Let T N be a random tiling of R N . Let G D be the domino group. Let ψ : G D → Z be a surjective homomorphism. As N → ∞, the real random variable converges in distribution to a normal distribution centered at 0. In other words, there exist C 0 , C 1 ∈ (0, +∞) with the property below.
This result can be considered a variant of the Central Limit Theorem. We prove it in Section 5 using characteristic functions: if X is a real valued random variable, set ϕ X (t) = E(exp(itX)).
Our final result is related but significantly different.
Figure 2 may count as experimental evidence in favor of Theorems 8 and 9. The proof of Theorem 9 is given in Section 6; it uses Theorem 8. It might at first seem that Theorem 9 would follow rather directly from Theorem 8, but that does not seem to be correct. The probability in the statement of Theorem 8 can be written as a summation containing roughly But it could happen, at least in principle, that some of the terms are much larger or much smaller than their neighbors.
An important example of surjective homomorphism is ψ = Tw : C D → Z provided D is not trivial: Theorem 3 is a corollary of Theorem 9, as claimed. Theorem 9 also implies a stronger version of the second claim in Corollary 2.2.
Theorems 7, 8 and 9 depend upon the structure of the domino group G D : we need either a subgroup or a homomorphism with certain properties. In this, they are more general than the results in the introduction, which are stated for regular disks only. There are applications of the stronger results to cases where D is not regular, and we saw a sample in Corollary 2.2. Theorem 4, however, assumes full knowledge of the domino group and is therefore not clear what could be said in greater generality.

Random tilings and proof of Theorem 7
It follows from the identification between tilings of corks and paths in the complex C D , discussed in the previous section, that any question concerning random tilings admits a translation as a question concerning random paths in C D . We try to give elementary proofs.
Lemma 3.1. Given a quadriculated region D ⊂ R 2 there exist λ 1 > 0, c ∈ (0, 1) and a unit vector v 1 ∈ R P with positive coordinates such that (when N → ∞) Proof. We can apply the Perron-Frobenius Theorem to the irreducible matrix A. Let λ 1 > 0 be the eigenvalue of largest absolute value, with associated unit eigenvector v 1 . We know that λ 1 is simple, that the coordinates of v 1 can be taken to be positive and that there exists c ∈ (0, 1) such that |λ j | < cλ 1 for any other eigenvalue λ j . Set Π = v 1 v * 1 , the orthogonal projection onto the line spanned by v 1 . Write A = λ 1 (Π + C) where ΠC = CΠ = 0 and C is a strong contraction: |Cv| < c|v| for all v ∈ R P {0}. We thus have The first claim is thus proved.
For the second claim, we have From the first claim, if j and N − j are large we have as desired.
We are almost ready to present the proof of Theorem 7. We present the proof for the right coset space {Kg; g ∈ G D }; the other case requires only minor adjustments.
Recall that for well behaved path-connected spaces with base point (Z, z 0 ) there exists a natural correspondence between subgroups K < π 1 (Z; z 0 ) and connected covering spaces Π : Z K → Z. Given K, the covering is characterized by the fact that the image of the induced map π 1 (Π) : π 1 (Z K ) → π 1 (Z) equals K. Also, the degree of the covering equals the index of K in π 1 (Z).
Proof of Theorem 7. Apply this construction in our case to define a covering space Π : C K D → C D of degree n. By the usual construction, C K D is a finite 2complex. Let P K be the set of vertices of C K D : we call these lifted plugs. The base point p K • ∈ P K of C K D is a fixed preimage under Π of the empty plug p • ∈ P. Tilings of R N can be lifted to paths of length N in C K D , starting at the base point p K • ∈ P K and ending at any preimage of p • . There exists a natural identification Π −1 [{p • }] ≈ G D /K: for a tiling t ∈ T (R N ), ψ(t) ∈ G D /K is the final point of the corresponding lifted path in C K D . The contruction in the proof of Lemma 3.1 (using the Perron-Frobenius theorem) can be applied to paths in the complex C K D . Moreover, the eigenvalue of largest absolute value is the same number λ 1 > 0 (for both C D and C K D ) and the corresponding eigenvector v K 1 is the lift of v 1 : for p ∈ P K we have for both i = 0 and i = 1. Thus for all p 0 ∈ G D /K, completing the proof.
The reader will recall from [7] that vertical floors are very useful in our constructions. Let vert(t) be the number of vertical floors of t. The following result shows that for almost all tilings vertical floors are relatively abundant. In the proof below the reader should keep in mind that we are not trying to give good estimates of the constants C and c.
Proof. Let p • = D ∈ P be the plug with all squares marked. Let v 1 ∈ R P be the unit vector with positive coordinates introduced in Lemma 3.1. Consider = 1 2 min p∈P (v 1 ) 2 p > 0. Again from Lemma 3.1, let N be such that for all j > N , N > j + N and for all p 0 , pÑ ∈ P and forT a random tiling of TÑ ;p 0 ,pÑ we have We imagine that T is created floor by floor (with the correct conditional probabilities). Let N j = jN . Assuming T constructed up to floor N j−1 we have Prob[plug N j (T) = p • | · · · ] > (where the dots stand for the description of the already constructed part of T). We thus have at worst N/N floors, each with conditional probability at least of being vertical. The claim now follows.

Cocycles
Let D be a nontrivial quadriculated disk. Let ψ : G D → Z be a surjective homomorphism so that ψ ∈ Hom(π 1 (C D ); Z). It is a well known fact that there exists a natural isomorphism between Hom(π 1 (C D ); Z) and the cohomology space H 1 (C D ) so that we may interpret ψ as an element of H 1 (C D ). We review this construction, which will be important for us in any case.
Let K < G D be the kernel of ψ. As reviewed in the previous section, let Π : C K D → C D be the covering space corresponding to K. The space C K D is an infinite (but locally finite) 2-complex; let P K be its set of vertices (plugs with an extra discrete information). The map ψ induces a natural correspondence between Z and the set of preimages (p (k) • ) k∈Z of the base point p • ∈ C D . Indeed, a tiling t ∈ T (R N ) defines a closed path in C D which lifts to a path in C K D from its base point p • . Notice that this definition is consistent: ψ(t 0 ) = ψ(t 1 ) if and only if ψ(t 0 * t −1 1 ) = 0 if and only if [t 0 * t −1 1 ] ∈ K if and only if the endpoints of the lifted paths to C K D coincide. More generally, there is an action of Z on C K D generated by the deck transformation σ : • . Recall that a deck transformation of the covering map Π : C K D → C D is a map Ψ : C K D → C K D satisfying Π•Ψ = Π. By construction, we have σ(p (k) (for all k ∈ Z). Letf 0 = (p 00 , f * 0 , p 01 ) and f 1 = (p 10 , f * 1 , p 11 ) be oriented edges of C K D so that p ij ∈ P K . If Π(f 0 ) = Π(f 1 ) then there exists k ∈ Z such thatf 1 = σ k (f 0 ) (so that, in particular, p 1j = σ k (p 0j )).
We construct a function ξ : P K → R such that, for every p ∈ P K and every k ∈ Z we have ξ(σ k (p)) − ξ(p) = k. Indeed, for each p ∈ P choose a Π-preimagẽ p ∈ P K and arbitrarily define ξ(p) ∈ R. Next, for all k ∈ Z define ξ(σ k (p)) = ξ(p) + k, completing the construction of ξ. Notice that iff 0 = (p 00 , f * 0 , p 01 ) and f 1 = (p 10 , f * 1 , p 11 ) are edges of C K D and satisfy Π(f 0 ) = Π(f 1 ) then ξ(p 01 )−ξ(p 00 ) = ξ(p 11 ) − ξ(p 10 ). For any oriented edge f of C D , define dξ(f ) = ξ(p 1 ) − ξ(p 0 ). The function dξ is well defined. In the language of cohomology, dξ ∈ Z 1 (C D ) ⊂ C 1 (C D ) is a cocycle and defines the same element of the cohomology space H 1 (C D ) as the original ψ under the natural isomorphism mentioned above. Also, if ξ 0 and ξ 1 are different functions obtained by the construction above then ξ 1 − ξ 0 is a well defined function from P to R so that dξ 1 −dξ 0 = d(ξ 1 −ξ 0 ) ∈ B 1 (C D ), consistently with the fact that they define the same element of the cohomology H 1 (C D ). The function ξ can be constructed so as to assume values in 1 m Z (for some positive integer m) or even in Z.
Alternatively, we know from Lemma 3.1 in [7] that for every plug p there exists a tiling t p ∈ T (R 0,2|D|;p•,p ): fix one such tiling for every plug p (recall that |D| ∈ 2N is the number of squares of D). Given a floor f = (p 0 , f * , p 1 ), let t f ∈ T (R 4|D|+1 ) be obtained by concatenating t p 0 , f and t −1 p 1 . We can then take ψ ξ (f ) = Tw(t f ) ∈ Z, thus defining a cocycle ψ ξ ∈ Z 1 (C D ) and the function ξ.
For either construction, if t ∈ T (R N ), t = (p 0 , f 1 , p 1 , . . . , f N , p N ) (with p 0 = p N = p • ) then This is consistent with the formulas for the twist seen, for instance, in [7].
Assume below for simplicity that ξ and dξ assume values in 1 m Z. Letq be a formal variable; let q =q m . Let Z[q,q −1 ] be the ring of Laurent polynomials in the variableq. We define the ψ-adjacency matrix or ξ-adjacency matrix α ∈ (Z[q,q −1 ]) P×P . For disjoint p,p ∈ P, let α (p,p) = f ∈D p,p q dξ((p,f,p)) = f ∈D p,pq (m dξ((p,f,p))) ∈ Z[q,q −1 ]; if p andp are not disjoint, define α (p,p) = 0. Alternatively, we may regard α as a function α : C {0} → C P×P where α(z) is obtained from α by substituting z forq; notice that α(1) = A ∈ Z P×P is the adjacency matrix of C D , as defined in Section 2. Also, The Laurent polynomial P N will be used again. An element P ∈ Z[q,q −1 ], P = 0, is a monomial if it has precisely one nonzero coefficient. If z ∈ S 1 ⊂ C, N ∈ N and p,p ∈ P then |((α(z)) N ) p,p | ≤ |(A N ) p,p |. Furthermore, there exists N 0 (depending on D and ψ only) such that if N ≥ N 0 then (α N ) p,p ∈ Z[q,q −1 ] is neither zero nor a monomial; moreover, there exists k ∈ Z such that the coefficients ofq k andq (k+m) are both positive. In this case, if z = exp( it m ) with t ∈ [−π, 0) ∪ (0, π] then the inequality is strict: Proof. The first claim (i.e., the non strict inequality) follows from the fact that (α N ) p,p is a Laurent polynomial with natural coefficients. For the second claim, since ψ is surjective, there exists even N 0 > 4|D| such that there exist tilings t 0 and t 1 of R 2|D|,N 0 −2|D|;p•,p• with ψ(t 1 ) − Tw(ψ 0 ) = 1. Use Lemma 3.1 in [7] to construct arbitrary tilings t a and t b of the corks R 0,2|D|;p,p• and R N 0 −2|D|,N ;p•,p , respectively. Concatenate t a , t i and t b to definet i ∈ T (R 0,N ;p,p ). The tilings t 0 andt 1 contribute withq k andq (k+m) to (α N ) (p,p) , respectively, proving that (α N ) (p,p) is neither zero nor a monomial. Moreover, for z as in the statement, |z k + z (k+m) | < 2, proving the strict inequality.
For the same D, ψ and N = 60, P 60 has terms from 673511306237603716 q −88 to 673511306237603716 q 88 ; the largest coefficients are of the order of 10 156 for q k , |k| ≤ 10 (i.e., k = Tw(t)), and are shown in Figure 2.
Our result can be considered a variant of the Central Limit Theorem and its proof will be similar. Let T N be a random tiling of R N (with uniform distribution); we also consider the integer valued random variable Ψ N = ψ(T N ). We then have here ϕ Ψ N is the characteristic function of the random variable Ψ N . We prove that these characteristic functions, with a minor adjustment, converge uniformly over compacts to a gaussian when N goes to infinity: this implies Theorem 8.
Lemma 5.1. Let D be a quadriculated disk and ψ : G D → Z be a surjective homomorphism; let η k : R → R, 1 ≤ k ≤ |P|, be the continuous maps above.
Proof. Let v 1 ∈ (0, 1) P ⊂ C P be the eigenvector of α(1) = A corresponding to λ 1 , as in Lemma 3.1. Consider the convex compact set Y ⊂ C P of all vectors y ∈ C P such that |y p | ≤ (v 1 ) p (for all p ∈ P). We claim that ( and N is sufficiently large; notice that this claim implies the first item. From the first claim of Lemma 4.2, if y ∈ Y and z = exp( it m ) then and therefore Y is invariant under (α(t)) N . From the second claim of Lemma 4.2, if N ≥ N 0 then some inequality in the display above is strict, completing the proof of the first item.
We already saw that η 1 is even, real analytic in a neighborhood of t = 0, and that η 1 (0) = 1 is a local maximum point. In order to complete the proof of the second item we are left with proving that η 1 (0) < 0. We again consider the invariant set Y . Take N ≥ N 0 so that, from Lemma 4.2, for all p,p ∈ P there exists k ∈ Z such that the coefficients ofq k andq (k+m) in (α N ) p,p are both positive. There exists thereforec > 0 and˜ > 0 such that, for all t ∈ (−˜ ,˜ ) and for all p,p ∈ P, we have |((α(t)) N ) p,p | ≤ (1 −ct 2 )((α(1)) N ) p,p . We thus have (α(t)) N [Y ] ⊆ (1 −ct 2 )Y and therefore, for all k, |η k (t)| N ≤ (1 −ct 2 ); taking k = 1 we obtain the second item. The third item is then easy.
where the error term o(1) goes to 0 exponentially (when N goes to +∞). Let the real function p 1 is even and real analytic with p 1 (0) = 1 (notice that the denominator is known to be positive from Lemma 3.1).
Recall (again from Lemma 3.1) that Thus, when N goes to infinity, here again the error term o(1) goes to 0 exponentially in N and uniformly in t ∈ (− , ). We now estimate the characteristic function Together with Equation (3) (in Lemma 5.1) we thus have or, more precisely: It follows that the sequence of characteristic functions (ϕ Ψ N / √ N ) N ∈N converges uniformly in compacts to the gaussian exp(−a 2 t 2 ), completing the proof.
Given a nontrivial quadriculated disk D, a surjective homomorphism ψ : G D → Z and K = ker(ψ), construct ξ : P K → R as in Section 4. For p 0 , p N ∈ P and a tiling t ∈ T (R 0,N ;p 0 ,p N ), interpret t as a path in C D and lift it to C K D , thus defining a finite sequence (p k ) 0≤k≤N of vertices of P K . Define Notice that if p 0 = p N = p • then ψ ξ (t) = ψ(t); in other words, we are extending the definition of ψ to tilings of corks.
Corollary 5.2. Consider a nontrivial quadriculated disk D and a surjective homomorphism ψ : G D → Z; construct ξ as above. Given two fixed plugs p,p ∈ P, let T be a random tiling of R 0,N ;p,p . As N → ∞, the real random variable (1/ √ N )ψ ξ (T) converges in distribution to a normal distribution centered at 0. Furthermore, the limit distribution does not depend on the choices of ξ, p orp.
Proof. The result follows from Theorem 8 together with Lemma 3.1. Alternatively, it follows from the proof of Theorem 8.

Proof of Theorem 9
Our aim is to prove Theorem 9 (using Theorem 8): it follows directly from Lemmas 6.1 and 6.3 below. The quadriculated disk D and the surjective homomorphism ψ : G D → Z will be fixed. The constants C 0 , C 1 ∈ (0, +∞) are as in Theorem 8. We shall first introduce some (local) notation.
It follows from the surjectivity of ψ that there exist N • ∈ N * and tilings t •,0 , t •,1 of R N• with ψ(t •,1 ) = 1 + ψ(t •,0 ). For the rest of the section, N • and t •,i will be fixed.
For a tiling t of R N , let B(t) be the set of k ∈ Z, 0 ≤ k ≤ (N/N • ) − 1, with the following properties: • the plugs p kN• and p (k+1)N• of t both equal p • ; • the restriction of t to R kN•,(k+1)N• is a translated copy of either t •,0 or t •,1 . Let Define also an equivalence relation ≡ in T (R N ) (not to be confused with ≈ or ∼). We have t 0 ≡ t 1 if and only if: Let t be a tiling, ψ(t) = τ . Let k i be the number of t •,i blocks in t, so that k 0 + k 1 = β(t). In the equivalence class [t], the random variable ψ(T) follows a binomial distribution: Lemma 6.1. Let D ⊂ R 2 be a quadriculated disk; let φ : G D → Z be a surjective homomorphism; let C 0 , C 1 ∈ (0, +∞) be as in Theorem 8. Let (t N ) be a sequence of integers with lim N →∞ t N / √ N = τ 0 ∈ R. We then have We first need a lemma about binomial numbers. We follow the convention n b = 0 if b < 0 or b > n. Lemma 6.2. For every > 0 there exist δ > 0 with the following property. Given δ ∈ (0, δ ) there exists n δ ∈ N such that if n > n δ , and a ∈ Z, then Furthermore, given a compact interval J ⊂ (0, δ ), the value of n δ can be taken to be the same for all δ ∈ K.
Proof. The inequality is trivial if a < 0 or a > n. Assuming 0 ≤ a ≤ n, notice first that and therefore ∆ > 0 for (n − 2a) 2 > n + 2. The letter ∆ is used to remind us of the Laplacian: ∆ is a discrete second derivative. The concavity of the graph of Assuming δ < 1 and n large, this takes care of the case |a − n 2 | ≥ 3 √ n.
For the case |a − n 2 | ≤ 3 √ n we also assume δ < 1 so that |b − n 2 | ≤ 4 √ n for all b in the summation. Stirling approximation formula yields assuming the conditions above on a and b, the approximation holds uniformly. Both this equation or the Central Limit Theorem yield assuming δ ∈ K, K ⊂ (0, 1) a fixed compact interval, this also holds uniformly. Set We have Given > 0, there exists δ > 0 such that the fraction on the right hand side lies in the interval (1 − 2 , 1 + 2 ) for all t 0 ∈ [−4, 4] and all δ ∈ (0, δ ), completing the proof of the lemma.
Proof of Lemma 6.1. Let (t N ) be a sequence of integers, as in the statement, with We prove that for any ∈ (0, 1 10 ) we have This will prove our lemma.
Given δ > 0, let τ ± = τ 0 ± δ. With τ 0 fixed, for sufficiently small δ > 0: Apply Theorem 8 to deduce that, for sufficiently large N , Let t 0 be a tiling of R N with β(t 0 ) > C • N . Recall from Equation (4) that Prob[ψ(T N ) = t | T N ≡ t 0 ] follows a binomial distribution. We may therefore apply Lemma 6.2 to deduce that Since this holds for any such t 0 , and since the total measure of the equivalence classes with β ≤ C • N is very small, we have Put together Equations (5), (6) and (7) to obtain the desired conclusion.
Lemma 6.3. Let D ⊂ R 2 be a quadriculated disk; let φ : G D → Z be a surjective homomorphism; let C 0 , C 1 ∈ (0, +∞) be as in Theorem 8. Let (t N ) be a sequence of integers with lim N →∞ t N / √ N = τ 0 ∈ R. We then have Proof. For tilings of R N , we consider the real variable τ = ψ(t)/ √ N ∈ R. We know from Theorem 8 that the random variable ψ(T)/ √ N converges in distribution to the gaussian g(τ ) = C 0 exp(−C 1 τ 2 ) in the τ -axis. If we restrict ourselves to an equivalence class [t] we have of course a binomial distribution. Letβ = β(t)/N andẽ(t) = e(t)/ √ N ; if we exclude a subset of T (R N ) of very small measure we may assume that C • ≤β ≤ 1/N • . The distribution of the random variable ψ(T)/ √ N in the equivalence class [t] thus also approaches (at least in distribution) a gaussian g t (τ ) = C 0,t exp(−C 1,t (τ −ẽ(t)) 2 ) in the τ -axis. Notice that the logarithmic derivative is an affine strictly decreasing function. The condition C • ≤β ≤ 1/N • implies C 1,min ≤ C 1,t ≤ C 1,max . Here C 1,min , C 1,max are constants: more precisely, they are functions of D only, not of N or of t. Thus, the slopes of the logarithmic derivatives g t /g t are uniformly bounded (excepting the set of very small measure which we are consistently neglecting).
For −2 ≤ i ≤ k, set τ i = τ 0 + iδ and J i = [τ i−1 , τ i ] ⊂ R. From Theorem 8, for sufficiently large N we have For such N , consider the equivalence classes [t] which satisfy C • N ≤ β(t) ≤ N/N • . Such an equivalence class [t] (and by extension, the tiling t) is of type I if g t (τ 1 )/g t (τ 0 ) ≤ λ and of type II otherwise.

Proof of Theorems 4, 5 and 6
The proof of Theorems 4, 5 and 6 now mostly amount to putting together previous results. We begin by (essentially) restating Lemma 11.1 from [7]. In order to prove uniqueness, consider a fat componentF with Tw(F) = t. By definition, there exists a tiling of the form t 1 * t vert,M ∈F. We have that t 1 is a tiling of R N −M with Tw(t 1 ) = Tw(t 0 ). Since D is regular, t 0 ∼ t 1 . It follows from Theorem 2 that t 0 * t vert,M ≈ t 1 * t vert,M and thereforeF = F N,t , as desired. The third item is similar.
If t belongs to a thin component, we have vert(t) < M . The estimate follows from Lemma 3.2.
Proof of Theorem 5. From Theorem 9, the probability that Tw(T 0 ) = Tw(T 1 ) is more than (1 − )C 2 0 /N (the probability that both twists equal 0). This is important because we will neglect much smaller probabilities.
Assume first that K = ker(Tw) < G + D is finite with k = |K|. Choose (and keep) N 0 ∈ 2N * and tilings t 0 , . . . , t k−1 of R N 0 representing the distinct elements of K. In particular, Tw(t i ) = 0 (for all i). A block in a tiling t is a translated copy of one of the t i so that t = t − * t i * t + . By imitating Lemma 3.2, the probability that a random tiling T does not include at least one block is o(c N ) (for some c ∈ (0, 1)) and therefore negligible. We define an equivalence relation: t ≡t if and only if t = t − * t i * t + ,t = t − * t j * t + , and t − admits no block. In other words, t ≡t if and only ift is otained from t by replacing the first block t i by some other block t j . With the exception of tilings which admit no block, all equivalence classes have size k. Notice that t ≡t implies Tw(t) = Tw(t).
Assume that T 0 has been selected first. In each ≡-equivalent class with Tw(·) = Tw(T 0 ) there exists precisely one value of T 1 with T 0 ∼ T 1 . We already know that the probability that T 0 ∼ T 1 and T 0 ≈ T 1 is negligible. The desired probability tends to 1/k, as claimed.
Assume now that K is infinite. Consider k ∈ N * and select N 0 and tilings t 0 , . . . , t k−1 of R N 0 representing k distinct elements of K. Define the equivalence relation and equivalence classes as above. In each equivalence class there exists at most one value of T 1 with T 0 ∼ T 1 . This implies that the lim sup of the desired probability is at most 1/k. Since this holds for any k, the limit exists and is equal to 0, as claimed.
We do not know if there exists a quadriculated disk D which is not regular but for which K is finite.
Proof of Theorem 6. Takeb = b − 4c. The tiling t 0 is constructed as follows. The first 4 (or 2) floors contain a copy of one of the left tilings in Figure 4 (the other dominoes being vertical). The remaining N − 4 floors contain a tiling in F N −4,t 0 . Notice that Theorem 4 (and the choice ofb) guarantee that such a tiling exists. The tiling t 1 is obtained by performing the trit indicated in Figure 4. The two resulting tilings are in F N,t 0 and F N,t 0 +1 , as desired; indeed, we may assume they have M vertical floors (where M is as in Theorem 2 and as in the definition of a fat component).  There appear to be examples where the giant connected component G under flips and trits is actually equal to T (R N ). This is true for the boxes [0, 3] 2 × [0, 2] and [0, 4] 3 . This is not true for the rather special region shown in Figure 6 (from [4]). It would be interesting to clarify when we have G = T (R N ).

Final remarks
The most obvious open question is Conjecture 1.1. In the introduction, it is formulated for boxes, but, if true, there are probably similar statements for regions of other shapes. Also, the set T (R) of tilings should have fat connected components F t ⊂ T (R) for different values of the twist: Tw(F t ) = t.
It would also be interesting to have a better understanding of the thin connected components, both in the case of cylinders and for more general regions. There is some experimental evidence which hints that they should be very small.
For cylinders, a better understanding of which quadriculated disks are regular would be very helpful. A study of the irregular cases would also be desirable.
The main results of this paper give us significant information concerning the distribution of Tw(T) for T a random tiling of a cylinder. Many natural questions remain unanswered, however. Is the sequence Prob[Tw(T) = t] unimodal?
The question which is easiest to formulate is whether flips and trits make T (R) connected for R a box.