Tiling by bars
by Martin Tassy
B.Sc., ENS Cachan 2007 M.A., Universit´ePierre et Marie Curie 2008
Adissertationsubmittedinpartialfulfillmentofthe requirements for the degree of Doctor of Philosophy in Mathematics at Brown University
PROVIDENCE, RHODE ISLAND
May 2014 c Copyright 2014 by Martin Tassy This dissertation by Martin Tassy is accepted in its present form by Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.
Date
Richard Kenyon, Ph.D., Advisor
Recommended to the Graduate Council
Date
Je↵rey Brock, Ph.D., Reader
Date
Richard Schwartz, Ph.D., Reader
Approved by the Graduate Council
Date
Peter Weber, Dean of the Graduate School
iii Vitae
Martin Tassy was born in Marseille, France. After finishing high-school at Lyc´ee de Provence, he entered classes pr´eparatoires successively at Lyce Thiers and Lyc´ee Sainte-Genevieve. In 2006, he was admitted at Ecole´ normal sup´erieure de Cachan. During his four years as a ´etudiant normalien, he received a Bachelor of Science in Mathematics, a Master of Mathematics in Probabilities from Universit´eRennes 1, a Master in Financial Mathematics from Universit´eParis 6 in 2008, and the french Aggr´egation of Mathematics. After spending one year as an academic guest at ETH Zurich to work on Random Interlacements, he entered the Brown University graduate school in 2010.
iv Acknowledgements
This work is a PhD thesis supervised by Richard Kenyon whom I thank for his tremendous assistance. I am extremely grateful to him for introducing me to the subject of tilings and supporting my educational endeavors, as well as for the numer- ous hours he spent patiently discussing and reviewing my ideas despite my challenges with communicating them. Many thanks to Scott She eld for his intellectual advice and whose own thesis on random surfaces has been greatly inspirational to my work. My appreciation to Tim Austin who introduced me to cocycles and for his guidance on questions related to ergodic theory. To the members of my thesis jury, Je↵Brock and Richard Schwartz, and to the sta↵ of the Brown Mathematics department, thank you for your time and consideration. AParticularthankstoDoreenPappas,whoalwaysdidherutmosttoassistmedur- ing the four years of this program. Thanks to Emanuel Zgraggen and Philippe Eichmann who patiently assisted me when I needed help with programming related problems. Thanks to Sara Maloni, Tonya Spano and Pellumb Kelmendi who kindly reviewed parts of this work. Finally, the deepest thanks to my family, especially to my parents, for their infinite love and support.
v Abstract of “ Tiling by bars” by Martin Tassy, Ph.D., Brown University, May 2014
We study combinatorial and probabilistic properties of tilings of the plane by m 1 ⇥ horizontal rectangles and n 1verticalrectanglesalsocalledtilingsbybars. ⇥
In Chapter 2 We give a new criterion for the tilability of regions based on the height function in the Conway group. As a consequence we are able to prove that the tilability or regions with bounded number of holes and such that the boundary of those holes describes a trivial word in the Conway group is decidable in polynomial time. In the second part of Chapter 2 we generalize results on local moves connectiv- ity known for simply connected regions by giving a criterion to decide under which conditions tilings of a torus are connected by local moves.
In Chapter 3 we discuss the di↵erences between the space of ergodic Gibbs mea- sures for tilings by dominos and the space of ergodic Gibbs measures for tilings by longer bars. Using notions from ergodic theory we prove that when a measure µ on tiling of the plane is ergodic then the image of the height function must µ -almost surely stay close to a single geodesic. We also propose a characterization of those ergodic Gibbs measures based on the generalization of the domino slope and on the support of the height function. Contents
Vitae ii
Acknowledgments iii
1 Introduction 1 1.1 Tilingsandheightfunctions ...... 3 1.2 Translation invariant Gibbs measures on tilings ...... 8
2 Tiling of the plane 12 2.1 Definition of tilings of the plane ...... 13 2.1.1 Tilings and their Conway tiling group ...... 14 2.1.2 Tilings by bars ...... 16 2.2 Local flips on tiling by rectangles ...... 34 2.2.1 Tilings of simply connected regions in Z2 ...... 34 2.2.2 A criterion for local move connectivity of tori tilings ...... 38
3 Translation invariant measures on tilings by bars 43 3.1 Gibbs measures on tiling by bars ...... 44 3.1.1 Definition and basic properties ...... 44 3.1.2 Translation invariant Gibbs measures on tilings ...... 46 3.1.3 Ergodic Gibbs measure ...... 49 3.1.4 Support of the height function on ergodic Gibbs measures . . 51 3.2 Surface tension ...... 56 3.2.1 definition of the surface tension relative to a specific word . . 56 3.2.2 Minimizers of the entropy exist and are Gibbs measures . . . . 62 3.3 Construction of invariant measures with a given slope ...... 64
4 Conclusion 70 4.1 Existence and characterization of the ergodic Gibbs measures . . . . . 71 4.2 Variational principle and unicity of ergodics Gibbs measures . . . . . 73
v A Construction of the specific free energy 76 A.1 Entropyofamesure ...... 77 A.2 Constructionofthespecificfreeenergy ...... 80
B Notations 83
vi List of Figures
1.0.1 An aztec diamond randomly tiled by 3 1rectangles...... 3 ⇥ 1.1.1 The Cayley graph of Z3 Z3 ...... 5 1.2.1 Random tilings of a torus⇤ by 2 1and3 1rectangles...... 9 ⇥ ⇥ 2.1.1 Local picture of the set S! ...... 27 1 3.3.1 Torus with null vertical slope ...... 66 3.3.2 Torus with Extremal slope ...... 66 3.3.3 Combined configuration ...... 67 3.3.4 Combined configuration after shu✏ing ...... 68
vii Chapter One
Introduction 2
Tilings as mathematical objects have been the subject of a countless literature over the last fifty years. In certain specific cases, such has tilings of the square lattice by dominos or tilings of the triangular lattice by hexagons, their properties are fairly well understood, and questions concerning their limiting behaviors or estimates of the number of tilings of a given region can be answered accurately (see amongst others [16],[4]). In other cases even the question of tilability of a region have been proven to be NP complete (see [18]).
For most of the models for which precise results have been obtained, a tiling can be seen as the projection onto the plane of a 3-dimensional surface. Hence there exists a natural function which associates to every vertex of the tiling a number corresponding to the height of this surface; this is the so called height function. In [6] Conway introduced a generalized notion of height function for which the image space is not necessarily Z but could be a specific group associated to the tiling. When this group has a complicated structure a few results have been proven but they are still far from reaching the precision of the results obtained for height functions in Z.
This thesis an attempt to understand better those generalized height functions by studying the specific case of tiling by bars. These tilings have proven to be simple to study because even though the image group of the height function is complex, there exist a Glauber dynamic (see [13]) which allows to randomize tilings of finite regions R and study their limiting properties (see Figure 1.0.1). 3
Figure 1.0.1: An aztec diamond randomly tiled by 3 1 rectangles ⇥ 1.1 Tilings and height functions
AprototileoftheplanelatticeZ2 is a connected shape which covers exactly a finite number of squares of the lattice. A tiling of a region R of the plane Z2 by a finite set of prototiles t ,...,t up to translation is a collection of prototiles such that { 1 k} each square of R is covered by one and only one tile. In [6] Conway and Lagarias introduced a useful tool to describe tilings of a given region. The idea was to associate awordinthefreegroupF = a, b to every path contained in the tiling. The letter a,b h i 1 a would represent an horizontal step to the right, a an horizontal step to the left, b
1 averticalstepupandb averticalstepdown.Ifwedenote!i’s the words described be the boundaries of the prototiles t ,...,t we can define the Conway group of the { 1 k} tiling to be G = a, b ! = ... = ! = e , h | 1 n i 4
It is easy to see that two paths contained in the tiling with same starting and ending points must have the same projection into G. Hence this construction provides a necessary condition for the tilability of simply connected region of the plane since such regions should have a boundary path projecting into the identity word in G. This result is useful in itself but it appears that for certain types of tilings the height function contains much more information. In fact for some special sets of prototiles, tilings of a region R can be identified with the set of k-Lipschitz functions (with certain properties) from R to the Conway Group. The first result of this kind was obtained by J.C Fournier in [7] for the domino model. In this article Fournier showed that a simply connected region R would be tilable by dominos if and only if we had for all x,y in the boundary of R:
d(h(x),h(y)) 2 x y +1. || ||1
However the domino model has a very special place in the field of tilings due to its correspondence with the dimer model and the simple structure of its Conway group which can be embedded to Z.ThepurposeofChapter2istoproveageneralization of those results for tilings by bars and provide a method that we believe could be apply to other type of tilings.
In the case of tiling by m 1horizontalbarsandn 1verticalbars,R.Kenyon ⇥ ⇥ and C.Kenyon proved in [13] that the Conway group can be naturally projected into
Zm Zn.WecanalsoassociateanaturaldistancedZm Zn to this projection which is ⇤ ⇤ minimal number of letters used to write an element of Zm Zn. ⇤
This projection is maximally informative in the sense that it contains all the information about the tiling. More precisely Remila showed in [22] that such tilings of a region R could be identified with functions in Zm Zn such that: ⇤ 5
Figure 1.1.1: The Cayley graph of Z3 Z3 ⇤
h(x0)=eZm Zn where x0 is the lower left corner of the region R. • ⇤
For every pair of points x =(x ,x )andy =(y ,y )inR,ifwewrite: • 1 2 1 2
1 i j h(x) h(y)= a k b k , k 0 Y
we have ik =(x1 y1)modm and jk =(x2 y2)modn. If this condition k 0 k 0 is verifiedP we will say that x and y satisfyP the modulus condition.
For each pair (x, y)ofneighborverticesinRwehave: •
dZm Zn (h(x),h(y)) 3 ⇤
For every neighbor vertices x, y in the boundary R¯ of the region: •
dZm Zn (h(x),h(y)) = 1 ⇤
Using this characterization of tilings, we show that tilability of a region can be reduced to a set of conditions on the height of the points in the boundary R¯.Infact 6 we give a more general statement which get rids of the simply connected assumption. This is the main theorem of this chapter.
Theorem. Let x ,...,x be a set of points in the plane together with a set of heights { 1 n} 2 g1,...,gn in Zm Zn. There exist a tiling of Z with height function h such that { } ⇤ h(x )=g ,...,h(x )=g if and only if g ,...,g respects the modulus conditions { 1 1 n n} { 1 n} and for any i, j n we have:
x1 x1 / (m ↵ (i, j)) , ↵ (i, j) j i 2 k k k 0 k 0 ! X X or x2 x2 / (m (i, j)) , (i, j) j i 2 k k k 0 k 0 ! X X ↵k(i,j) k(i,j) 1 where a b is the only way to write gi gj in Zm Zn such that for every k 0 ⇤ triplet k,Q i, j we have 1 ↵ (i, j) m 1 and 1 (i, j) n 1. k k
The proof of this theorem relies on the construction of extremal tilings associated to a starting point and an asymptotic direction given by boundary point of the 1 group Zm Zn. These extremal tilings possess two fundamental properties: ⇤
1. They push the height function as fast as possible towards the boundary point in any given direction of the plane 1
2. The height of any point is within distance 2 of a half-geodesic of the group
which join the trivial word eZm Zn to . ⇤ 1
We show that by intersecting such tilings with two di↵erent starting points x and y it is possible to construct a new tiling if and only if x and y verifies the conditions Theorem. A generalization of this argument to n point allows us to obtain our result. 7
One of the advantages of this theorem over the Fournier version for dominos is that it does not make use of the notion of boundary points. This allows us to construct the same criterion for tilability independently of the number of holes in the region R.Themostnoteworthyconsequenceofthisconstructionconcernsthe complexity of deciding tilability of non simply connected regions.
Corollary 1. The tilability of a region with a bounded number of tilable holes is decidable in polynomial time.
The second section of this chapter focuses on connectivity of tilings by local moves. It has been proven in [13] that any two tilings of a simply connected region can be connected by a sequence of local moves which consist in turning mn-vertical bars in nm-horizontal bars and vice versa. In the specific case of tiling by dominos, Saldanaha, Omei, Casarin Jr. and Romualdo proved in [24] that the same result holds on a torus provided that height di↵erence on the vertical and horizontal sections of the the torus for both tilings are the same. We prove the the analogous result in the
h v case of tiling by bars. Namely if we denote by !T and !T the words described by the vertical and horizontal paths around the torus for the tiling T ,wehave:
Theorem. For any couple of tilings T and T 0 of a torus such that there exist a
h h v v possible local move on T or T 0. If we have ! = ! and ! = ! then T and T 0 T T 0 T 0 T 0 are connected by local moves.
The main additional di culty compared to the simply connected case is that the height function is no longer well-defined. However, given a starting point with trivial height and under the conditions of this theorem it is still possible to define
1 the di↵erence of the height functions hT (x)hT 0 (x)foranypointx in the torus. Moreover this height function di↵erence remains the same if two points are in the same component of the clusters where the tilings agree. Using this observation and 8 the construction of maximal paths, which replaces global extremum for the height function, we are able to show that there is a necessarily a sequence of local moves that connect the two tilings.
1.2 Translation invariant Gibbs measures on tilings
Gibbs measures on tilings of the plane by a set of prototiles t ,...,t can be seen { 1 k} as a natural generalization of uniform measures on tilings of finite regions. In fact, given ⌦the space of tilings of Z2,wesaythatameasureµ is a Gibbs measure if for every finite connected region ⇤ Z2 we have: 2
Eµ [ ⇤c ]= ⇤c ( ) , ⇧|F ⇧
where ⇤c is the uniform measure on tilings of ⇤conditioned on the boundary tiles.
Since Gibbs measures are often studied as limiting measures of microscopical systems, it is natural to be interested in the set of translation invariant Gibbs mea- sures. This is a convex set whose extremal points are the ergodic Gibbs measures. Thus, if we are able to characterize completely what are the ergodic Gibbs measures are for a given type of tiling, we will have a full understanding of the translation invariant Gibbs measures on these tilings. One of the goals of this thesis is precisely to understand the set of ergodic Gibbs measures could be for tiling by bars.
In the specific case of domino tilings, Gibbs measures have been extensively studied and translation invariant Gibbs measures are now well understood. It has been proven by Kenyon, Okounkov and She eld in [16] and [26] that those ergodic Gibbs measures can be characterized uniquely by the expected variation of a one- 9 dimensional height function. In order to extend these results to the case of Bars, a natural approach would be to use multidimensional abelian height functions obtained by assigning a weight to edges depending on the type of squares they separate in a m n chessboard coloring of the plane as described in [25]. However, this approach ⇥ does not work as I am going to explain below.
Since ergodic Gibbs measures satisfy a zero-one law on the sigma-algebra of translation invariant events, a natural way to grasp a first intuition of the set of ergodic Gibbs measure is to construct translation invariant measures and look for the macroscopic characteristics of the picture. For tiling by bars there exists the Glauber dynamics introduced in Chapter 1 which randomize tilings of finite regions. We used this dynamics in order to simulate random tilings of tori such as the ones in Figure 1.2.1 and obtain translation invariant measures.
Figure 1.2.1: Random tilings of a torus by 2 1 and 3 1 rectangles ⇥ ⇥
The two pictures in Figure 1.2.1 have very similar appearance at first glance. However, upon closer examination, we see that for 3 1tilingswehavemultiples ⇥ choices in the pattern of the frozen red region, which is not the case for the domino tiling. Moreover, the pattern cannot be described by a commutative height function as it was expected since this pattern does not depend solely on the numbers of edges of each types described by the border of the dominos but also by the order in 10 which they appear. This means that, for max(m, n) 3, characterizing translation invariant Gibbs measures through a commutative height function cannot be su cient to describe the set of ergodic Gibbs measures entirely. This is why we need the full height function on Zm Zn. ⇤
As mentioned previously, tilings of the plane by bars can be seen as Lipschitz
2 functions from Z to the hyperbolic group Zm Zn.Suchobjectshavebeenstudied ⇤ by people working in ergodic theory.
Given G y X be an action of a group G on a set ⌦and given another group L ,a cocycle for the action of G on L is a map a : G X L such that for all , ⇥ ! 1 2 2 and all ! ⌦wehave 2
↵ ( 1 2,x)=↵ ( 2,x) ↵ ( 1, 2x) .
If there exists a measure µ on ⌦such that the action of on µ is ergodic we say that ↵ is an ergodic cocycle.Fromthisdefinitionweobtainthatforanyergodic Gibbs measure on tilings on the plane, the set of height functions define an ergodic cocycle from Z2 to the Conway group of the tiling. When this group is hyperbolic, as
Zm Zn,thisimposessomestrongrestrictionsontheimagesetoftheheightfunction ⇤ and we have the following result:
Theorem. Let µ be an ergodic Gibbs measure on Z2. Then for µ-almost every tiling
2 T there exists a geodesic ray ! such that for any vector v R we have: 2
1 lim dZm Zn hT (x kv ), ! =0, k ⇤ b c !1 k 2 where h x k ,! is the height of the closest point of k in Z and dZm Zn is the b c b c ⇤ distance in the Cayley graph of Zm Zn. ⇤ 11
The proof uses the same kind of technique as the Karlsson-Ledrappier theorem in [11]. Applying the subadditive ergodic theorem, we can define, for any ergodic measure µ,asemi-norminR2 which encodes the asymptotic speed at which the height function is travelling on the Cayley graph of Zm Zn in any given direction. ⇤ The translation invariance of the measure imposes that the set X of the zeros of this semi-norm can only be an hyperplane. In particular the complement Xc of this set has exactly two connected components. Now, if we denote v,! the half- geodesic which supports the height function in the direction v,onecanprovethat the function v is almost surely continuous on Xc for the discrete topology, and 7! v,! thus constant on each component of Xc.Thismeansthat,foranygivenconfiguration !,theimageoftheheightfunctionissupportedonexactlytwohalfgeodesicswhich together form a single geodesic of Zm Zn. Another consequence of this theorem ⇤ is that the zeros of the height function have a preferred direction in Zm Zn which ⇤ corresponds to the direction of the white paths in Figure 1.2.1.
Morever the construction of this semi-norm tells us that the limits
1 sh =lim dZm Zn ((0,n), 1Zm Zn )) n ⇤ ⇤ !1 n and 1 sv =lim dZm Zn ((n, 0), 1Zm Zn )) n ⇤ ⇤ !1 n both exists and are µ-a.s surely constants. Those limits should be seen as the anal- ogous for tiling by bars of the slope in the domino model.
It is now natural to attempt to construct ergodic Gibbs measures by fixing a periodic geodesic and an asymptotic vertical and horizontal speed. In the last section of Chapter 3 we describe a qualitative construction of those Ergodic Gibbs measures. Chapter Two
Tiling of the plane 13
In this chapter we will be interested in tilings of the plane and their height functions as introduced by Conway and Lagarias. The main result we want to emphasize is that for certain class of tilings, the height function contains somehow all the information on the tilings. This means that height functions do not only provide a necessary condition on the tilability of a region as described in [6] but one can also obtain a su cient condition for tilability based on the distance of the height of the boundary points in the Conway group. We will cite several consequences of this result for the tiaibility of subsets of the plane.
The second section of this chapter deals with local move connectivity for regions of the planes. We extend the result first obtain my R. Kenyon and C.Kenyon in [13] to the case of tilings of a torus.
2.1 Definition of tilings of the plane
Throughout the rest of this thesis we will always identify the plane with Z2.A prototile t of the plane is a simply connected figure in the square lattice of Z2.
2 For any region R in Z and any finite set of prototiles t1,...,tk ,atilingofR by { } t ,...,t is a collection of prototiles such that each square of R is covered by one { 1 k} and only one tile. If there exist such a tiling of R by t ,...,t ,wesaythatthe { 1 k} region R is tileable by t ,...,t . A tiling problem consist of a region R and a finite { 1 k} set of tiles. 14
2.1.1 Tilings and their Conway tiling group
1 Let t ,...,t be a set of tiles. Let a be an horizontal step on the right and a be { 1 k} 1 an horizontal step on the left. Equivalently, let b be a vertical step up and b be a vertical step down.
Definition 2. afinitesequencesofverticesp =(v ,...,v )isapathifforeveryi n 0 n we have v v 1Ifp is a path we can define a path word ! (p)=u ...u k i i+1k1 0 n 2 1 1 F = a, b ,suchthatoreveryi n such that u ⌃= a, a ,b,b represents a,b h i i 2 { } the step to go from vi to vi+1.
To every tile ti and any arbitrary chosen starting point on the tile, we can asso- ciate a word ! in the free group F = a, b corresponding to the path described by i a,b h i the perimeter of the tile starting at this point. In [6] Conway and Lagarias introduced agroupassociatedeverysetofprototiles.
Definition 3. The Conway group of the tiling t ,...,t is the quotient group { 1 k}
G = a, b ! = ... = ! = e . h | 1 n i
We will also use the notation CG to mention the Cayley graph of G
The first remark me can make is that the definition of the group G does not depend on the starting point we choose for each !i. Indeed, two tiles words which describe the same prototile boundary but with di↵erent starting points are conjugate with each other. Thus one has a trivial representation in G if and only if the other one has also a trivial representation.
The first property of this group is that it provides a necessary condition for 15 tilability of a simply connected region R.
Lemma 4. Let R be a connected region with boundary R¯, if R is tilable then we have projG(!R¯ =1G) with !R¯ being the word in Fa,b associated to the boundary path and projG being the projection to the Conway group G
Proof. The proof is realized by induction on the number n of tiles of a tiling T of R.Thecasewheren =1followsfromthedefinitionofG and by definition one does not change the the boundary word of a given tiling by removing a tile on the boundary.
Now given a finite connected region R,thelowerleftcornerx0 of R and a tiling T of this region, we can associate to T the function f : R G in the following R R TR,x0 ! way:
f (x )=e where e is the trivial word in G. • 0
If x, y is a horizontal right edge (resp. vertical down edge) in T ,wehave • { } R f (y)=f (x) a (resp. f (y)=f (x) b).
When there is no ambiguity in the region R and in the starting point we will use
fT for fTR,x0 .
This tells us that the function fTR is well defined since two connected paths with the same starting and ending points go around a tiling.
Remark. The tiling function is defined for simply connected regions but can natu- rally be extended to regions with holes such that each hole is tilable. 16
f (x)=baaba
It is important to notice that this is just a necessary condition and in general even if the projection of any closed path to G is trivial, this does not necessarily mean that any simple connected region is tilable. For example if we try to tile the figure below by 3 1rectangles,itisstraightforwardtocheckthattheboundarywordis ⇥ trivial in the Conway group but the region is not tilable. This happens because one can add vertical tiles on the corner of the region R to obtain a tilable 3 4rectangle ⇥ without changing the boundary word. In fact the triviality of the boundary word means that there exist a finite sequence of move of adding and removing tiles which ends in the empty set.
2.1.2 Tilings by bars
We will now be specifically interested in the case of tilings by m 1-rectanglesand ⇥ a n 1-rectangleswhichwewillcalltilingbybarsfromnowon.Thiscasehasbeen ⇥ primarily studied by C. Kenyon and R. Kenyon in [13] and subsequently by Remila 17
3 2 1 1 2 p (f ¯)=bab a b a b 3a =1 G R G in [1] and [22]. In the rest of this section we will mention several results from those papers.
m m 1 n n 1 In the case of tiling by bars we have !1 = a ba b and !2 = b ab a ,and thus the corresponding Conway group becomes
m m 1 n n 1 G = a, b a ba b = b ab a = e . h | i
Due to its complicated structure, this groups is hard to work with. In order to simplify the problem we can look for projection into groups with structure which can be more easily manipulated. One option could consist in using an abelian height function as suggested in She eld [25]. An e cient mean to construct such a height function is to use 3-color chessboard coloring of Z2 and count how many times a path is winding left or right of squares with a given color. However the rest of this chapter shows that such a height function cannot capture the full complexity of the space of tiling and this is why we will use a more complicated non-abelian height function. 18
We consider the proposition introduced by Kenyon and Kenyon in [13] which consist in projecting this group in
m n Zm Zn = a, b a = b = e . ⇤ h | i
Even though this group is not commutative, Its structure is much easier to under- stand than the one of the original Conway group and its Cayley graph is very similar to a tree as shown in the picture below.
3 3 H = a, b a = b = e = Z3 Z3 | ⇤ ⌦ ↵ One can notice from Remila [22] that this is not the only relevant projection however we will prove that it is a maximally informative one in a certain way.
An important property of tilings by bar is that they are “local tilings” in the sense that there exist a set of local conditions to verify on the height functions which are su cient to assure that the function describe a tiling of a region R.Inother words if the height function satisfies those conditions everywhere then the global height function represents a unique tiling. This has been proved by Remila in [22] 19 but for the sake of completeness we will recall and prove this theorem.
Theorem 5. For any g1,g2 in Zm Zn, let dZm Zn (g1,g2) be the minimal number of ⇤ ⇤ 1 letter used to write the word g1 g2 in Zm Zn. Let R be a simply connected region ⇤ of the plane, x0 be its lower left corner and h be a function from R to Zm Zn = ⇤ a, b am = bn = e such that we have: h | i
h(x0)=1Zm Zn • ⇤
For every pair of points x =(x ,x ) and y =(y ,y ) in R, if we write: • 1 2 1 2
1 i j h(x) h(y)= a k b k , k 0 Y
we have ik =(x1 y1)mod[m] and jk =(x2 y2)mod[n] If this k 0 k 0 conditionP is verifies we will say that x and yPsatisfy the modulus condition.
For each pair (x, y) of neighbors vertices in R we have: •
dZm Zn (h(x),h(y)) 3 ⇤
where dZm Zn is the distance in the Cayley graph for the set of generators a, b . ⇤ { }
For each pair (x, y) of neighbors vertices in the boundary R¯ we have: •
dZm Zn (h(x),h(y)) = 1 ⇤
Then there exist a unique partial tiling T of R such that h = hT
Proof. We begin by proving the following claim: 20
Claim: If x and y are neighboring vertices with y = x+(0, 1) (resp. y = x+(1, 0)) and dZm Zn (x, y) 3 then there exist j with 0 j n 1(resp.iwith0 i m 1) ⇤ j j i i such that h(y)=h(x)b ab (resp. h(y)=h(x)a ba ).
Moreover for every j0 such that j n j0 j (resp. i0 s.t i m i0 i)wehave
j0 h(x +(0,j0)) = h(x)b and
j0 h(y +(0,j0)) = h(y)b
i0 i0 (resp. h(x +(i0, 0)) = h(x)a and h(y +(i0, 0)) = h(y)a ).
This first part of the claim comes from the observation that we must have
i1 i2 i3 dZm Zn (x, y)=1ordZm Zn (x, y)=3.IndeedIfwewriteh(y)=h(x)b a b ⇤ ⇤ (resp. h(y)=h(x)aj1 bj2 aj3 ), the modulus condition tells us that we must have
i1 i1 i1 + i3 =0modm and i2 =0modn.Thuswecanwritethath(y)=h(x)b ab
j1 j1 (resp. h(y)=h(x)a ba )whichgivesusthefirstpartoftheproof.
We prove the second part of the claim for the case where y = x +(0, 1), the case y = x +(1, 0) follows directly by symmetry. Let x0 = x +(1, 0), y0 = y +(1, 0), u =
1 1 h(x) h(x0)andv = h(y) h(y0). By assumption we know that dZm Zn (x0,y0) 3. ⇤ 1 j j This means that we must have l(u b ab v) 3. Since the first part of the claim k k l l tells us that there exist k and l such that u = a ba and u = a ba ,Itisdirectto check that this is only possible if u = b and v = b.
This gives us the proof for j0 =1.Byrepeatingthisargumentsuccessivelyfor x0 = x +(0,j0)whilej0 j in one direction and j0 n j in the other direction, we obtain the complete statement. 21
Me now have enough elements to define the tiling T associated to the height function h.Wesayshatthatacelliscoveredbyanhorizontaltile(resp.verticaltile) if there is a vertex x on the boundary of the cell such that h(x+(1, 0)) = h(x)bjab j with 0 The claim above guarantee that there is no overlap and that each horizontal (resp. vertical) tile is of length m (resp. n). Now suppose that we have x1,x2,x3,x4 are the four vertices on the boundary of a cell labelled in the clockwise direction with x1 in the lower left corner, and that we have dZm Zn (h(xi),h(xi+1)) = 1 for ⇤ 1 1 i 3. This implies that we must have h(x )=h(x )aba b which is contradictory. 1 1 Consequently there is at least one i 3 such that dZm Zn (h(xi),h(xi+1)) = 3 and ⇤ every cell is a covered by a tile of T .ThefactthathT = h follows from our construction concludes our proof The next theorem is the core of this section. it gives a necessary and su cient condition for the existence of a tiling with fixed height function at a given set of points. Theorem 6. Let x ,...,x be a set of point in the plane together with a set of height { 1 n} 2 g1,...,gn in Zm Zn. There exist a tiling of Z with heights h(x1)=g1,...,h(xn)= { } ⇤ { g if and only if g ,...,g respects the modulus conditions and for any i, j n we n} { 1 n} have: x1 x1 / (m ↵ (i, j)) , ↵ (i, j) j i 2 k k k 0 k 0 ! X X or x2 x2 / (m (i, j)) , (i, j) j i 2 k k k 0 k 0 ! X X ↵k(i,j) k(i,j) 1 where a b is the only way to write of gi gj in Zm Zn such that for k 0 ⇤ every tripletQ k, i, j we have 1 ↵ (i, j) m 1 and 1 (i, j) n 1. We call k k 22 this way to write the minimal writing of the element. This theorem has a lot of applications for tilability of regions. Before we start to prove it, we will give here several interesting consequences of this result Theorem 7. Let R be a finite region of the plane with boundary R¯ and such that the height function is well defined on the boundary R¯. Then R is tileable if and only if for all x =(x1,x2) and y =(y1,y2) in R¯ the following conditions holds: y1 x1 / (m ↵ (x, y)) , ↵ (x, y) 2 k k k 0 k 0 ! X X or y2 x2 / (m (x, y)) , (x, y) 2 k k k 0 k 0 ! X X ↵k(x,y) k(x,y) 1 where a b is the minimal way to write h(x) h(y) in Zm Zn k 0 ⇤ Q Proof. This follows directly from Theorem 6 by taking x ,...,x to be the set of { 1 n} boundary points. Another noteworthy consequence of this theorem is the following corollary: Corollary 8. Let R be a finite region of the plane with boundary R¯ and such that the height function is well defined on the boundary R¯. If for all x, y R¯ we have: 2 dZm Zn (h(x),h(y)) max(m, n) x y ⇤ k k1 then R is tilable. Proof. This is just a simple consequence of Theorem 6 and the fact that ↵ m and k n k 23 We can also apply Theorem 6 to the tilability of Regions of the planes with holes. Indeed we can use the same type of demonstration to give a criterion to tile such regions Theorem 9. Let R be a region of the plane with k-holes and boundary R¯ and such ¯ that every component of R describe the trivial word in Zm Zn. R is tilable if and only ⇤ if there is a set of height on those component which verifies the modulus condition and such that we have or all x, y R¯: 2 y1 x1 / (m ↵ (x, y)) , ↵ (x, y) 2 k k k 0 k 0 ! X X or y2 x2 / (m (x, y)) , (x, y) 2 k k k 0 k 0 ! X X ↵k(x,y) k(x,y) 1 where a b is the minimal way to write of h(x) h(y) in Zm Zn. k 0 ⇤ Q Proof. The proof of this theorem is also a straight application of the method used for the proof of Theorem 6. Corollary 10. The tilability of a region with a bounded number of holes and such that all the boundary of the hole describe the trivial word in Zm Zn is decidable in ⇤ polynomial time. Proof. Let M be an upper bound of the number of holes. We first Notice that, according to the last theorem, if we know the height on the border of every hole then the problem becomes necessarily polynomial. Indeed we have to test the condition of Theorem 6 on R¯ points which has complicatedity O( R¯ 2 ). Moreover for every | | | | | connected component of R¯, if we fixed the height of one point, we can compute the height of all the other point in at most O( R¯ ). | | 24 So that the only exponential complexity comes from the set of height we have to try for a fixed point on the boundary of the holes. The key observation comes with the following claim: ¯ Claim: Let Ci i n be the di↵erent connected components of R ,IfRistilable { } then there exist a tiling T such that i nBZm Zn (hT (Ci) , 4) is simply connected. [ ⇤ Let us consider the case where there exist a tiling T 0 such that i nBZm Zn (hT (Ci) , 4) [ ⇤ 0 form two connected components of CZm Zn that we denote 1 and 2.Wecanal- ⇤ C C ways choose two points x1 and x2 such that dZm Zn (hT (x1),hT (x2)) = dZm Zn ( 1, 2). ⇤ 0 0 ⇤ C C Moreover due to the tree structure of CZm Zn ,wecanfindapointg with dZm Zn (x1,g) ⇤ ⇤ 4 which satisfies the same modulus condition as hT (x2)andsuchthatifwewrite 1 ↵k k hT (x1)g = a b Yk and 1 ↵l l hT (x1)hT (x2)= a b Yl we have k ↵k < l ↵l and k k < l l . P P P P 1 Suppose that we consider the height h on R¯ such that for all x h ( )wehave 2 C1 1 1 h (x)=h (x)andforally h ( )wehaveh(y)=gh (x ) gh (y). For any T T 0 2 C2 T 0 2 T 0 two point x and y in R¯ in the same we have Ci 1 1 h(x) h(y)=hT 0 (x) hT 0 (y) 25 If x and y then 2C1 2C2 1 1 1 h (x)h(y)=hT (x)ghT (x2)hT (y) But since the geodesic path between hT (x) hT (y)goesthroughhT (x2)wemusthave 1 1 1 l(hT (x)hT (y)) = l(hT (x)T (x2)) + l(hT (x2)hT (y)) and 1 1 1 1 l(h (x)gh (x )h (y)) l(h (x)g)+l(h (x )h (y)) T T 2 T T T 2 T Since by assumption we know that k ↵k < l ↵l and k k < l l,bycombining the two inequalities we obtain thatP the conditionsP of TheoremP 5 mustP still be verified by the height function h. Consequently Corollary 10 tells us that there exist a tiling ¯ T such that hT = h on R and since dZm Zn (x1,g) 4, the set i nBZm Zn (hT (Ci) , 4) ⇤ [ ⇤ is simply connected. In the general case where the heights on R¯ from n connected components. We can always split the components into two groups with disjoints convex hulls and use the same argument to prove that there is a tiling with a height function that bring those two convex hulls closer. This proves our claim. Now every connected component of the boundary contains at most N 2 points so that the Ball of radius 4 around each of this component contains at most 24N 2 points. By looking at the number of way to identify two points of a component This gives us a maximum of (24N 2)2M possible connected component. And thus the complexityy is polynomial We are now getting back to the proof of Theorem 6 26 To prove this theorem we must first introduce new objects coming from the theory of Gromov hyperbolic groups which will be used in the demonstration. Definition 11. Let be a point in the boundary @(Zm Zn)ofZm Zn.Forany 1 ⇤ ⇤ fixed point x in Zm Zn, there exist a geodesic segment from g to to which we ⇤ 1 can associate an infinite infinite word !g, representing this segment. Informally, 1 the infinite word !g, represent the straightest path from g to the boundary point 1 1 From now on we will not distinguish the geodesic segment from the infinite word !g, which represents it. 1 Definition 12. Let !g, be an infinite infinite word in @(Zm Zn). We define the 1 ⇤ extremal set S!g, associated to !g, as the set of h in Zm Zn such that there 1 1 ⇤ is a subword ! of !g, with length greater than 2 and dZm Zn (h, !) 2. We also 1 ⇤ defined the geodesic distance d which is the distance associated to the boundary 1 point with the convention that d (eZm Zn )=0 1 1 ⇤ When there is no ambiguity we will now use the notation ! for !g, and for 1 1 the following definitions we will consider g and fixed. 1 We define now straight paths between two points. This concept will be essential in several demonstrations of this chapter. ik jk Definition 13. Let u = a b be an element of Fa,b representing a path in the k l plane and ! an infinite wordQ of Zm Zn.Wesaythatu is an ! -straight path if 1 ⇤ 1 ik jk for all k l we have ik m, jk n and a b is a subword of ! . | | | | k l 1 1 Q We define now the extremal tiling associated to the infinite word ! . This is 1 the tiling obtained while trying to go as fast as possible in the asymptotic direction 27 given by .Onecouldthinkofitasthenaturalanalogofminimalandmaximal 1 tilings in the domino case. As we have proved it in Theorem 5, a tiling is uniquely defined by its height function, it is thus su cient to describe a tiling by its height at every point of the plane and show that this height function verifies every condition of Theorem 5. Definition 14. Let x be a point of the plane and g and be fixed. For all y Z2 1 2 we define h! (y)the! -height of y as the only gy S! such that 1 1 2 d! (gy)= min d! (g!), 1 ! P! ,y 1 2 1 where P! ,y is the set of ! -straight paths starting at x and ending at y. 1 1 Figure 2.1.1: Local picture of the set S! 1 This allow us to define the extremal tiling of the plane T! ,x as the tiling such 1 2 that for all y Z we have h(y)=h! (y).Weneedtoprovethatthistilingiswell 2 1 defined. 2 Theorem 15. T! ,x is a tiling of Z 1 Proof. To prove this theorem we need to show that for every x in Z2 the height 28 h! (x)iswelldefinedi.ethatforeveryx there exist at least one ! -straight path 1 1 ending at y.Wealsoneedtoshowthatthefunctionh! (x)verifiesallthecondition 1 of 5 Clearly we do not lose any generality by restricting ourselves to the case where the base point is (0, 0) Z2 and where ! starts with an horizontal step a.Suppose 2 1 ik jk that x =(x1,x2)and! = a b ,wedefinebyinductionthefollowingpath: 1 k 0 Q If we have i m x i and x =0thenwedefineux = ux = ax1 • 1 1 1 2 1 Let xn be the ending point of the path ux,if x xn m and x xn n • n | 1 1 | | 2 2 | n n x x x1 x1 x2 x2 x then u = una b otherwise if un is ending in a (resp. b), we define ux = uxa↵n+1 (resp. ux = ux b n+1 )where↵ = i if x xn and n+1 n n+1 n+1 n+1 n 1 1 ↵ = m i otherwise (resp. = j if x xn and = n j n+1 n n+1 n 2 2 n+1 n otherwise). This process if finite since x xn m implies x xn+1 m and x xn m | i i | | i i | | i i | implies x xn x xn+1 . | i i || i i | We need to show that the function verifies the conditions of Theorem 5. It is straight forward that the modulus condition must be verified for all x since h! (x) 1 can be written as the projection in Zm Zn of a path from the origin to x and ⇤ this projection does not change the abcissa modulus m (resp. ordinate modulus n) because all the boundary words of prototiles are closed paths. Moreover we can make the following observations. For any two element g1,g2 which verifies the same modulus condition, we have dZm Zn (g1,g2) 4. ⇤ 29 This comes from the fact that ker (projmod[m,n](Zm Zn)) is the group generated by ⇤ i j i j a b a b i m,j n. We do not recall the demonstration of this claim here but it { } can be found in Section 3.3 of Remila [22]. In particular this tells us that there exist a unique element in S! with the correct 1 modulus condition which minimize the geodesic distance to ! . Hence the function 1 h! is necessarily well defined. Finally for every point g in S! it is not hard to check that at least one element 1 i i j j of ga ba i m (resp. gb ab j n)isalsoinS! . This can be seen in Figure 2.1.1. { } { } 1 This means that any neighbor of x has a representant h(y)in! such that there 1 exist a path ending at y with word h(y)anddZm Zn (h!(x),h(y)) 3. Consequently ⇤ by the definition of h! ,wemustnecessarilyhavedZm Zn (h! (x),h! (y)) 3. This 1 ⇤ 1 1 finishes the proof that T! ,x0 is a tiling. 1 In fact the algorithm given in the proof of Theorem 15 gives us exactly h! (x) 1 for any x in Z2.Byconstructionofthealgorithmweuseeithertheminimalnumber of steps to reach the abscissa of x or the minimal number of steps to reach this its while staying in S! .Thisassurethatthereisnopathp0,x staying in S! ,starting 1 1 at the origin and ending in x such that d! (projZm Zn (p0,x)) d! (h! (x)). 1 ⇤ 1 1 We have now everything we need to prove our main result. 30 Extremal tiling in the direction a2b2a2b2a2b2... Proof of Theorem 6. Let be a chosen point of the boundary of Zm Zn.Forevery 1 ⇤ i n we can construct a maximal tiling Txi,gi, in the direction and such that 1 1 2 the height of xi is gi by setting for every x in Z : hTx ,g , (x)=gihT 1 (x) i i xi,g 1 i 1 1 where gi in the infinite word going from gi to .Wecannowdefinetheextremal 1 1 tiling T = Txi, by 1 i n 1 T hT (y)=hTx ,g , (y) 1 i(y) i(y) 1 where i(y)issuchthatwehave d (h (y)) = max d (h (y)) Txi(y),gi(y),! Txi,gi, 1 1 i n 1 1 We claim that h (y)isuniquelydefineandthatT is a tiling. Throughout Txi,gi, 1 1 the rest of this proof we suppose the xi’s fixed as well as (y)sothatwecandenote 1 31 h = h for all i n. i Txi, 1 Claim: h (y)isuniquelydefined. Txi,gi, 1 Let i and j be two points such that we have d (hi(y)) = d (hj(y)) 1 1 1 There is a unique point gint(i,j) which is such thatt such that gi gint(i,j) is both a 1 1 subword of gi gj and gi g .Inothertermsthisisthepointwherethetwogeodesics 1 1 1 gi g and gj g meet. We will use the notation 1 1 1 ↵k k gi gint(i,j) = a b k n Y i and 1 ↵l l gj gint(i,j) = a b l n Y j were both way to write are minimal writing. By definition we necessarily have 1 ↵k k ↵l l gi gj = a b a b k n l n Y i Y j in Zm Zn According to our construction of extremal tilings described in Theorem ⇤ 15, if we define Ri to be the rectangle with lower left corner x1 (3 ↵ ),x2 (3 ) i k i k k n k n ! X i X i and upper left corner 1 2 xi + ↵k,xi + k k n k n ! X i X i 32 and Rj to be the rectangle with lower left corner x1 (3 ↵ ),x2 (3 ) 0 j k j k 1 k n k n X j X j @ A and upper left corner x1 + ↵ ,x2 + , 0 j k j k1 k n k n X j X j @ A we have that for any point x in @R Rc (resp. @R Rc), i [ i j [ j d (hi(x)) d (gint(i,j)) 1 1 respectively d (hj(x)) d (gint(i,j)). 1 1 If we look at the sets S 1 and S 1 this means actually that hi(x)(resp. hj(x)) gi g gj g 1 1 is in S 1 .Thereisnowtwocasetobeconsidered. gint (i,j)g 1 c c If y (@Ri Ri ) (@Rj Rj), hi(y)andhj(x)arebothonS 1 .Inthis gint (i,j)g • 2 [ \ [ 1 case we can use the same argument we used in the construction of extremal tilings and say that there exist a unique g(y)inS 1 such that gint (i,j)g 1 d (hi(y)) = d (hj(y)) = d (g(y)) 1 1 1 and thus hi(y)=hi(y). If y @R Rc since the interior of R and R are disjoints we have y/@R Rc • 2 i [ i i j 2 j [ j and d (hi(y)) which gives us a contradiction. The same applies to y @R Rc by symmetry. 2 j [ j 33 2 So that finally hT (y)iswelldefinedforeveryy Z and T is a tiling 1 2 1 2 Suppose now that there exist a tiling T of Z such that h(x1)=g1,...,h(xn)=gn . { } We claim that for any x, y mz2 we must have: 2 y1 x1 / (m ↵ (x, y)) , ↵ (x, y) 2 k k k 0 k 0 ! X X and y2 x2 / (m (x, y)) , (x, y) 2 k k k 0 k 0 ! X X ↵k(x,y) k(x,y) 1 where a b is the minimal way to write h(x) h(y)inZm Zn.Wewill k 0 ⇤ prove thisQ claim by induction on the distance dZm Zn (hT (x),hT (y)). If dZm Zn (hT (x),hT (y)) ⇤ ⇤ the result is straight forward to check and the claim is true. If dZm Zn (hT (x),hT (y)) ⇤ 2 we can always choose a point z such that z [x ,y ] • 1 2 1 1 z [x ,y ] • 2 2 2 2 According to our induction hypothesis we must necessarily have 1 1 (z x ) / (m ↵k(x, z)) , ↵k(x, z) • 2 k 0 k 0 ! P P 1 1 (z x ) / (m ↵k(x, z)) , ↵k(x, z) • 2 k 0 k 0 ! P P 1 1 (y z ) / (m ↵k(z,y)) , ↵k(z,y) • 2 k 0 k 0 ! P P 1 1 (y z ) / (m ↵k(z,y)) , ↵k(z,y) • 2 k 0 k 0 ! P P 34 But due to the supperaditivity of distance in the Cayley graph we also have ↵ (x, z)+ ↵ (z,y) ↵ (x, y) k k k k 0 k 0 k 0 X X X and (x, z)+ (z,y) (x, y) k k k k 0 k 0 k 0 X X X So that the inequalities y1 x1 / (m ↵ (x, y)) , ↵ (x, y) 2 k k k 0 k 0 ! X X and y2 x2 / (m (x, y)) , (x, y) 2 k k k 0 k 0 ! X X are necessarily true. This concludes our proof. 2.2 Local flips on tiling by rectangles 2.2.1 Tilings of simply connected regions in Z2 In this section we will show the result from R. Kenyon and Kenyon [13] stating that tilings of a simply connected region are connected by local moves. We will also proves that this result can be extended to torus tilings under certain conditions. Definition 16. Let T be the tiling of a simply connected region R.Wesaythat T 0 is connected to T by local flips if there is a finite sequence of tiling T1,....,Tn such that T = T , T 0 = T and for every 1 i n 1, there exist a m n- 1 2 ⇥ rectangle Ri such that Ti and Ti+1 agree outside of Ri and Ti is tiled by m vertical 35 n 1-rectangles in R (resp. n horizontal m 1-rectangles) and T is tilled by n ⇥ i ⇥ i+1 horizontal m 1-rectangles (resp. m vertical n 1-rectangles). ⇥ ⇥ a local move for tilings by 4 1 horizontal bars and 3 1 vertical bars ⇥ ⇥ In order to prove this theorems we introduce the notions of distance between tilings Definition 17. Let g in Zm Zn we will use l(!)todenotethedistancefromg to ⇤ the origin in the Cayley graph CZm Zn .Itisalsothenumberoflettersintheshortest ⇤ way to write !. We define the distance between two tilings of R as: 1 R(T1,T2)= l(hT1 (x)hT2 (x)). x R X2 We can now define maximal vertices. 36 Definition 18. Amaximalvertexx for a pair of tilings (T,T0)isavertexsuchthat: h (x) = h (x) • T1 6 T2 1 sup l(h (x),l(hT (x)) is maximal with the previous condition • T1 2 It is important to notice that if two tilings are di↵erent there is necessarily a maximal vertice and that we must have for any maximal vertice x: 1 sup l(h (x),l(hT (x)) 2. T1 2 This follows form the following inequalities 1 4 l(h (x)hT (x)) l(hT (x)) + l(hT (x)) T1 2 1 2 and l(h (x)) + l(h (x)) 2sup(l(h (x),l(h (x))) T1 T2 T1 T2 We can now state and prove our theorem on local move connectivity. Theorem 19. Let T and T 0 be two tilings of a simply connected region R, T and T 0 are connected by local moves. Proof. The idea of the proof is to show that there is always a local move which decrease the distance between the two tilings. For that we consider a maximal vertex x, we can assume without losing any generality that l(h (x) l(h (x)). T1 T2 We want to prove that this vertex is inside a rectangle where is it possible to do a decreasing local move. The proof is for the case when there exist i, j > 0suchthat i j i j hT1 (x)=ua b but it works the same if hT1 (x)endswithb a by symmetry. 37 i j ( ( Let x0 = x +(1, 0), we can write h (x0)=ua b b l)ab l)with0 l Since l(hT1 (x)) is maximal, this impose the condition l + j = n and thus we can i+1 ( rewrite hT1 (x0)=ua b j). In a same way if we denote x0 = x +(0, 1) we have i j k k i j+1 h (x00)=ua b a ba with 0 k We can repeat this argument by induction for every point in the m n-rectangle ⇥ R with lower left corner r = x0 = x +(i, j). This gives us that for any k m and 0 0 l n k l hT1 (r0 +(k, l)) = ua b This is exactly saying that R0 is tiled in T1 by m vertical rectangles. If we apply a local move at R0 we obtain a tilling Tflip such that any point of the rectangle has a height of the from l k hTflip(r0 +(k, l)) = ub a . Since u ends in b by assumption, this means that for every k m and l n: hTflip(r0 +(k, l)) l k And thus the module condition impose that either we have hT2 (r0 +(k, l)) = ub a and 1 hT ((r0 +(k, l))hT2 ((r0 +(k, l)) = 1Zm Zn 1 ⇤ or 1 1 l(h ((r +(k, l))h ((r +(k, l))) R(Tflip,T2) < R(T1,T2). By applying successively this moves we end up on two tilings such that there is no maximal vertices. This is only possible if they are the same tiling and this complete our proof. 2.2.2 A criterion for local move connectivity of tori tilings We will now show an analogous result for the case of torus. This result has been shown for dominos in [24], it states that two tilings that are not extremal and have same height di↵erence along path of homotopy (0, 1) and (1, 0) are connected by local moves. In order to show this result we must first characterize tilings by the word obtained while reading the horizontal and vertical sections. Definition 20. Let T be a tiling of a k k-torus , we denote by !T and !T the two ⇥ h v words described by the tiling following the paths from the (0, 0) to (0,n)andfrom (0, 0) to (n, 0) It is interesting to notice that those words are invariant by m-translation in the horizontal direction and n-translation in the vertical direction. Another meaningful T T observation is that we necessarily have that !h and !v commute with each others. Indeed the word obtained by going around the torus following the path (0, 0), (0,n), (n, n), (n, 0), (0, 0) { } is T T T 1 T 1 !v !h (!v ) (!h ) . 39 And this word must be equal to the identity in Zm Zn since it is the boundary word ⇤ of a the tiling of a simply connected region. Moreover since the structure of Zm Zn ⇤ is the same as a tree, this means that there exist an irreducible word !I such that T sh T sv we can write !h = !I and !v = !I . As we will see in the next chapter the two coe cient sh and sh can be understood as asymptotic speeds of a tiling sometimes called slope. The parameter !I is a new parameter specific to tilings with longer bars than dominos and which could be interpreted as the ”language” in which the tiling is written Definition 21. We say the tiling of a n n torus is extremal if there is no possible ⇥ local move on this tiling Proposition 22. If T is not an extremal tiling then there is no straight path with non null homotopy in T Proof. Suppose that there exist a straight path p with non nul homotopy (↵, )and ik jk associated word !p = a b . According to our precedent remark, any translated k 0 of this path describedQ the exact same word. Now If there exist a local move T we can find a translated of this path that goes through a m n vertical or horizontal ⇥ rectangle. In particular it is possible to shorten the way to write this path in Zm Zn. ⇤ But since we supposed that the path is straight we must necessarily have i ↵n k k 0 X and j n. k k 0 X This is a contradiction. 40 Theorem 23. Let T and T 0 be two non-extremal tilings of a torus. If we have h h v v ! = ! and ! = ! then T and T 0 are connected by local moves. T T 0 T 0 T 0 Proof. The idea of the proof is similar to the one we used for the result on simply connected domains. However in this case the height function is not well defined since the height change between two points will depend on the homotopy of the path one ↵, uses to link those two points. Because of that we use hT (x)todenotetheword obtained while following a path of homotopy (↵, )betweentheoriginandx.The second main di culty lies in the fact that there in no global maximum for the height function on a torus which is the characterization we used to pick a maximal vertex and apply a local move. However we can make the following observations: If we fix one point as the base point, namely (0, 0), then the height di↵erence • between the two tilings is still well defined. Indeed when going around a path of homotopy (↵, )between0andx we find that ↵, 1 ↵, 0,0 1 h ↵ v 1 h ↵ v 0,0 (h (x)) (h (x)) = (h (x)) ((! ) (! ) ) ((! ) (! ) )(h (x)) T T 0 T T T T T T 0 0,0 1 0,0 =(h (x)) (h (x)) T T 0 (2.2.1) h v because !T and !T commute as remarked earlier. If we superpose the two tilings T and T 0 and we look at the component where • they agree, then we do not change the height di↵erence by picking a starting point x which is in same connected component as the origin. This means that the notion of distance between two tilings remains valid and our 41 goal will be to find local moves that strictly decrease the distance between T and T 0. For that we need to modify slightly the definition of extremal vertices. Atorus-maximalvertexx for a pair of tilings (T,T0)isavertexsuchthat: 0,0 1 0,0 (hT (x)) (hT (x)) = eZm Zn • 0 6 ⇤ There exist a point x on the boundary of the connected component containing • 0 (0, 0) where T and T 0 agree such that x is at the end of a straight path of maximal length starting at x0.Thismeansthatforanypointy on a straight path starting at x0,wehave: 0,0 1 0,0 0,0 1 0,0 l (h (x )) h (y) (h (x )) h (x) T 0 T T 0 T The second notion is well-defined because we assumed that T and T 0 were not T 0 extremal tilings. As for the proof of 19 we will suppose that we can write !x0,x = uaibj. Since the argument only uses the fact that x was a local extremum, the proof remains in fact exactly the same and we find that the m n-rectangle R with lower ⇥ 0 left corner x (i, j)istiledinT by m vertical rectangles Now if denote by T the 1 flip tiling obtained after a local move at R0,wefindthat 0,0 1 0,0 0,0 1 j i (h (x)) h (x)=(h (x)) ub a T Tflip T and since u is necessarily ending in b,thisgivesus: 0,0 1 0,0 0,0 1 0,0 l((h (x)) h (x)) 0,0 1 0,0 unless hT (x) u =1Zm Zn which is contradictory because hT (x)isinthesame ⇤ 42 module class as h0,0(x)) but u is not by assumption. In the end we have: T 0 R(Tflip,T) < R(T 0,T) which complete our proof since there is always a torus-extremal vertex unless both tilings agree on the whole torus. Chapter Three Translation invariant measures on tilings by bars 44 Is this chapter we will be interested in limiting behaviors of tiling by bars. More explicitly we will look at translation invariant Gibbs measure on tilings by bars which are the measure one expect to observe when rescaling a figure with increasingly smaller meshes. The first part of this chapter is dedicated to the definition of the main notion and concepts we will use. 3.1 Gibbs measures on tiling by bars 3.1.1 Definition and basic properties In this section we define Gibbs measures on tilings and recall some of their basic properties Throughout this chapter we will identify tilings of a region R with the R subset of (Zm Zn) which corresponds to admissible condition of Theorem 5: ⇤ h(x0)=1Zm Zn . • ⇤ For every pair of points x =(x ,x ), y =(y ,y )inR,ifwewrite: • 1 2 1 2 1 i j h(x) h(y)= a k b k , k 0 Y we have ik =(x1 y1)modm and jk =(x2 y2)modn k 0 k 0 P P For each pair (x, y)ofneighborsverticesinR we have: • dZm Zn (h(x),h(y)) 3. ⇤ 45 For each pair (x, y)ofneighborsverticesintheboundaryofR we have: • dZm Zn (h(x),h(y)) = 1. ⇤ 2 Let ⌦be the set of function from Z to Zm Zn.Wedenoteby the Borel sigma ⇤ F 2 algebra of the product topology on ⌦. Let S be a subset of Z ,wedenoteby S the F sigma-algebra with respect to which h(x)ismeasurableforallx S.Asubsetof⌦ 2 2 which belongs to some S with S Z is called cylinder set. F ⇢⇢ Definition 24. A measure µ on is a Gibbs measure if we have for every finite F region R of Z2: Eµ[ Rc ]=⇤Rc ( ) ·|F · where ⇤Rc (.)istheuniformmeasureontilingsofR conditioned on !Rc .Wedenote by the set of Gibbs measure on tilings of the plane. G This definition shows why Gibbs measures should be considered as the extension of the uniform measure to infinite regions of the plans. In particular, just like the uniform measure is the maximizer of this entropy amongst measure on tilings of a finite region, Gibbs measure can be seen as maximizers of the entropy in a sense that we will precise later. It is easy to see that the set of Gibbs measures is a convex set. Hence, in order to describe completely , it would be su cient to characterize the extreme point of G this set. Since Gibbs measure all have the same conditional probabilities on finite set it is natural to expect those those extreme point are in fact characterized by their limiting behavior. The following definition and theorem explicit this intuition. Definition 25. Let A ,WesaythatA is a tail event if for every finite region R 2F of the plane A c .TheSigmaalgebrageneratedbytaileventsiscalledthetail 2FR 46 sigma algebra and we denote it . I This allow us to give the following classic characterization of extremal Gibbs measures: Theorem 26. A Gibbs measure is extreme in if and only if µ is trivial on the G sigma algebra . I AproofofthisclassictheoremcanbefoundinChapter7of[8].Theideaofthe proof is in every points similar to the one of Theorem 28 given below. 3.1.2 Translation invariant Gibbs measures on tilings We will now be specifically interested in the set of translation invariant Gibbs mea- sures on tilings. Since we are interested in tilings by m 1horizontalbarsandn 1 ⇥ ⇥ vertical bars, it is natural to specify the group of translation accordingly. Let ⌧ be the group of translations generated by x x+(0,m)andx x+(n, o), ! ! and let ⌧ by the sigma algebra generated by translation invariant events; this is the F smallest sigma algebra containing the sets of the form h h(y) h(x) where { | 2E} 2 x, y Z and is a measurable set for the discrete topology in Zm Zn.Agradient 2 E ⇤ ⌧ Gibbs measure is the same as a Gibbs measure except that we replace c by c . FR FR Definition 27. We say that measure µ is ⌧-invariant if for all A and all ⌧ ⌧ 2F k,l 2 we have : µ (A)=⌧k,l(µ)(A) we denote by (⌦, ⌧ )thesetof⌧-invariant probability measures P F 47 It is clear that (⌦, ⌧ )isaconvexsetsinceforanyµ ,µ (⌦, ⌧ )wehave: P F 1 2 2P F ⌧k,l(µ1)+⌧k,l(µ2)=⌧k,l(µ1 + µ2) The following theorem gives a characterization of the extreme points of this convex set. Theorem 28. A probability measure on µ (⌦, ⌧ ) is extreme in (⌦, ⌧ ) if 2P F P F and only if µ is trivial on the sigma algebra ⌧ . F Proof. Suppose that there exist a set A in ⌧ such that we have 0 <µ(A) < 1. F Then we can define the two conditional probabilities: µ = µ ( A) A ·| µ⌦ A = µ ( ⌦ A) . \ ·| \ Moreover we have µA = µ⌦ A and we can write 6 \ µ = µA + µ⌦ A. \ but since the two measures can we written 1A µA = µA 1⌦ A µ⌦ A = \ µ \ µ⌦ A \ and A is translation invariant, the two measures are both in (⌦, ⌧ )andµ is not P F extreme. Conversely suppose that µ is trivial on (⌦, ⌧ )andcanbewrittenasan P F 48 average of two translation invariant probability measures: µ = µ +(1 )µ 1 2 with 0 < <1. Then µ1 is absolutely continuous with respect to µ and there exist afunctionf such that we can write µ = fµ.Wewillprovethatf is actually ⌧ 1 F measurable. For any c>0 we have: Eµ1 1 f Since f1 f c c1 f c ,usingthelastexpressionwemusthave { } { } Eµ1 1 f Eµ1 (f c)1 f 1 f c ⌧ =1f c 1 f c ⌧ +1 f and 1 f c =1f c ⌧µ-a.s. So that f c is translation invariant and f is ⌧ { } { } { } F 49 mesurable. Going back to our measures, the triviality of µ implies f = µ(f)=1µ-a.s. F⌧ and µ1 = µ.ThisconcludestheproofoftheTheorem. 3.1.3 Ergodic Gibbs measure In the next paragraphs we define ergodic measures on tilings which will be the focus of our work for the remaining of this chapter and we will explain why they provide a complete description of the set of translation invariant events. Definition 29. A ⌧-translation invariant mesure µ is ergodic if it is trivial on the translation invariant sigma algebra Ergodic Gibbs measures satisfy the following classical ergodic theorem. Theorem 30. Let µ be an a ⌧-ergodic measure, and let ⇤n n 0 be a sequence of { } cube such that ⇤ as n . For any µ-integrable function we have: | n|! 1 !1 1 lim f ⌧k,l = Eµ [f] n ⇤n !1 k,l ⇤n | | X2 This allow us to give an alternate definition of ergodic Gibbs measures which uses a mixing property. Theorem 31. A mesure µ in (⌦, ⌧ ) is ⌧-ergodic if and only if for every A ⌧ P F 2F and every sequence of cube ⇤n n 1 we have: { } 1 lim sup ⇤n µ(A ⌧k,lB) µ(A)µ(B) =0 n B | | \ !1 2F k,l ⇤n X2 50 Proof. Suppose that (31) is verified, then for ant translation invariant event A we have µ(A ⌧ A)=µ(A)sothatnecessarilyµ(A)=µ(A)2 and µ(A)=0or \ k,l µ(A)=1.Thismeansexactlythatµ is trivial on ⌧ and thus ergodic. Conversely, F if µ is ergodic then we can write: 1 1 ⇤n µ(A ⌧k,lB) µ(A)µ(B) = dµ ⇤n 1A ⌧k,l µ(A) | | \ B | | k,l ⇤n Z k,l ⇤n X2 X2 1 E µ ⇤n 1A ⌧k,l µ(A) | | " k,l ⇤n # X2 According to the ergodic theorem, the last term of the equation tends to zero when n goes to and thus (3.1.1) is verified. 1 We denote by the set of ⌧-translation invariant Gibbs measures. Clearly is G⌧ G⌧ aconvexset.Wehavethefollowingresult: Theorem 32. A gibbs measure µ is extreme in if and only if µ is ergodic 2G⌧ G⌧ and thus ex µ = µ (⌦, ⌧ ) 2G⌧ 2G⌧ \P F Proof. The proof is exactly the same as the one of Proposition 31. One can prove that if µ is an extreme point of µ then it it necessarily trivial on the ⌧-invariant 2G⌧ sigma algebra This prove that every translation invariant Gibbs measure is an average of ergodic Gibbs measures. In fact one can endow the space of ergodic Gibbs measures with a probabilities measure which describe this weighted average. Precisely we have: Theorem 33. For every translation invariant Gibbs measure µ, there exist a unique 51 measure on ex µ such that: Wµ 2G⌧ µ = ⌫ µ(⌫) ex µ ⌧ W Z 2G 3.1.4 Support of the height function on ergodic Gibbs mea- sures In this section we will show that when µ is an ergodic Gibbs measures then µ-a.s the height function of a tiling must stay asymptotically close to a single geodesic. This is to be expected since the hyperbolicity of the Conway group for tiling by bars and the fact that the height function is 3-Lipschitz imposes that every two parallel paths in Z2 are supported on the same geodesic. Using di↵erent applications of the subadditive ergodic theorem we give a sense to this intuition. We introduce the notion of cocycle coming from ergodic theory and which will prove to be useful for the study of ergodic Gibbs measures on tilings Definition 34. Given G y X be an action of a group G on a set ⌦and given another group L ,acocycle for the action of G on L is a map a : G X L such that for ⇥ ! all , andall! ⌦wehave 1 2 2 2 ↵ ( 1 2,x)=↵ ( 2,x) ↵ ( 1, 2x) . If there exists a measure µ on ⌦such that the action of on µ is ergodic we say that ↵ is an ergodic cocycle. Now if we consider ⌦the set of tilings of the plane with trivial height at the origin. For any ergodic measure on tilings, we can define an ergodic cocycle in the 52 following way. We consider the action ⌧ of Z2 on ⌦given by: ⌧ ((↵, ),T)=T⌧(↵, ) 1 where hT⌧(↵, ) (x1,x2)=hT (↵m, n)hT (↵m + x1, n+ x2). 2 2 And we can define an ergodic cocycle from Z (⌦ Z )toZm Zn: ⇥ ⇥ ⇤ ((↵, ), (T,(x1,x2))) = hT⌧(↵, ) (mx1,nx2) This also gives us a random pulled-back pseudo metric: DT (x, y)=dZm Zn (hT (x),hT (y)) ⇤ Using the language of cocycles, Karlsson and Ledrappier proved in [11] that for any asymptotic direction in the plane µ–a.e tilings stay close to a half geodesic in Zm Zn. ⇤ In Lemma 35 and Theorem 36 we will precise and strengthen this result. Lemma 35. Let be a vector in Z2 then the limit 1 s =lim DT (0,k ) k !1 k exists and is µ-a.e the same. Moreover the function N : s is a semi-norm in ! R2 2 Proof. The cocycle identity gives for (x1,x2)and(y1,y2)inZ 53 hT (mx1,nx2)= ((x1,x2),T) = ((y ,x ),T) (x y ,x y ),T 1 2 1 1 2 2 ⌧(y1,y2) = hT (mx1,nx2)hT (m(x1 y1),n(x2 y2)) . ⌧(y1,y2) So that substituting from above we get: DT ((mx1,nx2), (my1,ny2)) = dZm Zn hT⌧(y ,y ) (m(x1 y1),n(x2 y2)) ,eZm Zn ⇤ 1 2 ⇤ ⇣ ⌘ = DT ((m(x1 y1),n(x2 y2)) , (0, 0)) ⌧(y1,y2) Now define fT (x1,x2)=DT ((mx1,nx2), (0, 0)) Since DT is a pseudo metric, it satisfies the triangle inequality: D ((mx ,nx ), (0, 0)) D ((mx ,nx ), (my ,ny )) + D ((my ,ny ), (0, 0)) T 1 2 T 1 2 1 2 T 1 2 Combining the two last inequalities it becomes fT (x1,x2) fT (x1 y1,x2 y2)+fT (y1,y2)(3.1.1) ⌧(y1,y2) 2 Using (3.1.1), if we fix a vector =( 1, 2)inR we can now apply the Kingman’s subadditive ergodic theorem which tells us that: 1 s (T )= lim fT (k 1m, k 2n) k !1 k 54 exists and that this limits must satisfy s (T )=s (T⌧ ). Moreover it also clearly satisfies s↵ (T )=↵s (T )bylinearity. We will now show that the ergodicity of the system impose that this function is actually µ-a.e constant. Let and 0 be two arbitrary directions in Z2,ifweapply (3.1.1) two times we get: 0 0 0 0 fT (km 1,kn 2) fT (km 1 m 1,kn 2 n 2)+fT (m 1,n 2) ⌧( 10 , 20 ) 0 0 fT ( m 1, n 2)+fT (km 1,kn 2) ⌧(k 0 ,k 0 ) ⌧( 0 , 0 ) 1 1 2 1 1 2 0 0 +fT (m 1,n 2) Dividing by k and letting k going to infinity, the Lipschitz assumption gives us 1 the first and the last terms of the equations (3.1.2) are O( k ). The middle term converges to give sT ( ) sT ( ). ⌧( 0 ) Applying this with 0 replaced by 0 , we find that in fact sT ( )=sT ( ). ⌧( 0 ) for every . Since the action ⌧ is ergodic, the limit doesn’t depend on T ,sowemay write it s( ). Finally, having done this, by using (3.1.1) with vectors k and k 0 and going to the limit, we find that s satisfies the triangle inequality. Using standard argument we can extend those property to Q2 and then R2.Sothatfinally 2 2 s is linear, symmetric and satisfies the triangular inequality which concludes our proof We can now use this Lemma to prove that µ-almost surely a tiling is supported 55 on a single geodesic. Theorem 36. Let µ be an ergodic Gibbs measure on Z2. Then for µ-almost every 2 tiling T there exists a geodesic ray ! such that for any vector v R we have: 2 1 lim dZm Zn hT (x kv ), ! =0, k ⇤ b c !1 k 2 where x k ,! is the height of the closest point of k in Z and dZm Zn is the distance b c b c ⇤ in the Cayley graph of Zm Zn. ⇤ Proof. Let be such that s > 0. Since the height function is 3-Lipschitz, for every 0 B( , s we can write. 2 4 D (0,k 0 ) >D(0,k )+3 0 T T | | 3 >D(0,k )+ s T 4 s > . 4 This means that we can choose a countable open cover of = : s > 0 such D { } that on each balls we have s > 0 µ-almost surely. And thus s > 0 µ-almost surely on . D For a fixed tiling T since the space Zm Zn is Gromov hyperbolic, we can define a ⇤ function from ⌦ T to th boundary of Zm Zn such that (T, )istheasymptotic ⇥D ⇤ direction of T in the direction in Z2.Moreoverweknowthatforanytwoasymptotic direction the distance between them grows linearly. Using the same argument as in the previous paragraph this means that (T,.) is continuous for the discrete topology 56 on the boundary of Zm Zn.Inparticular (T,.)mustbeconstantonveryconnected ⇤ component of . Now since s is a semi-norm we know that is either a point, a line D D or the entire plane. If is the entire plane then the height function stay close to D the origin in every direction. If is a line we obtain picture such as those presented D in the introduction where level sets of the height function cross Z2 in the same direction. Finally is a point then the function s reaches its minimum on D ! S1.Thismeansthattheheightfunctionisgoingawayformtheoriginatapositive speed and the function d ( ) almost surely reaches its minimum on a finite set T, · in Z2. Thus it induces a translation invariant measure of set in Z2 which is almost surely finite and non-empty which cannot happen. 3.2 Surface tension 3.2.1 definition of the surface tension relative to a specific word In the previous section we have proven that is if a measure µ on tiling is ergodic then µ-almost every tilings stayed asymptotically close to a unique geodesic. Moreover through the construction of the semi-norm s we showed the existence of the limits 1 sh =lim dZm Zn ((0,n), 1Zm Zn )) n ⇤ ⇤ !1 n and 1 sv =lim dZm Zn ((n, 0), 1Zm Zn )) n ⇤ ⇤ !1 n 57 In the domino tiling case those limits becomes the expected di↵erence of the variation of the height function between two points, sometimes called slope. Consequently it is natural to attempt to construct an analogous of the surface tension based on this limits. However since we need to precise on which geodesic tilings are supported, it is necessary to add an additional parameter associated to this set of geodesics in Zm Zn.InthecaseofanergodicGibbsmeasuresupportedonauniqueperiodic ⇤ geodesic, it is su cient to precise the period of this geodesic. Before we explicit the construction of the surface tension, we prove a slightly stronger version of the subadditive ergodic theorem which will allow us to consider average sums over Z instead of average sums over N. Proposition 37. Let be the set of intervals in Z, µ be a ⌧-ergodic mesure and f⇤ :⌦ R be a family of functions satisfying for all a b c Z: ! 2 1 There exist M< such that sup ⇤ f⇤ Mµ-a.e • 1 ⇤ | | 2 f f + f µ-a.e. • [a,c] [a,b] [b,c] Then if we write 1 ↵ =liminf Eµ[f⇤] ⇤ ⇤ 2 | | we have: 1 lim f[ n,n] = ↵µ-a.e n !1 n Proof. For ✏>0, we can choose [↵ , ] suchthat: ✏ ✏ 2 1 Eµ[f⇤✏ ] ↵ + ✏ ⇤ | | ✏ 58 2 Then for every n N there exist two sequences (a(n),b(n)) N and ( 1(n), 2(n)) 2 2 2 2 such that for all n we have (n) ↵ and | i || ✏ ✏| [ n, n]= (n) [↵ a(n)( ↵ ),↵ b(n)( ↵ )] (n) 1 [ ✏ ✏ ✏ ✏ ✏ ✏ [ 2 Using the subbaditivity of the family (f⇤) we are allowed to write 2 f[ n,n] f 1(n) + f[↵✏ a(n)( ✏ ↵✏),↵✏ b(n)( ✏ ↵✏)] + f 2(n) f[↵✏ a(n)( ✏ ↵✏),↵✏ b(n)( ✏ ↵✏)] +2M ✏ ↵✏ | | This gives b(n) 1 (b(n) a(n))(b✏ a✏) 1 f[ n,n] f[a✏,b✏] ⌧(k) 2n 2n (b(n) a(n))(b✏ a✏) k=Xa(n) 2M⇤ + ✏ 2n The first term of the equation tends to Eµ[f[a✏,b✏]]bytheergodictheoremandthe second term tends to 0 so that we have lim f[ n,n] ↵ + ✏µ-a.e n !1 Since it remains true for all ✏,thisfinishestheproof. 59 In the following paragraphs we explicit the construction which allow us to define the surface tension associated to a periodic geodesic in Zm Zn ⇤ Definition 38. Let ↵0 be a word in Zm Zn and x and y be two points in Z,we ⇤ define the function p↵0, x,y going from ⌦to Z by { } 1 p↵0, x,y (T )= max l ( ): is a subword of both ↵0 and hT (x) hT (y) { } Zm Zn 2 ⇤ Definition 39. Let ↵ be a finite word in Zm Zn and let µ be a translation invariant ⇤ Gibbs measure on (⌦, ), we define the horizontal and vertical speeds Sh and Sv of F ↵ ↵ µ relatively to the word ↵ by h 1 S↵ (µ)= lim ⇤n Eµ p↵,( n,0),(n,0) n !1 | | ⇥ ⇤ v 1 S↵ (µ)= lim ⇤n Eµ p↵,(0, n),(0,n) n !1 | | ⇥ ⇤ Before we explicit the construction of the surface tension, we recall a classic result on subadditive functions which proof is very similar to the proof of the previous proposition. Lemma 40. Let ⇤ be the set of intervals in Z and f be a function from ⇤ to R which satisfies the conditions 1. For any ⇤ ⇤ we have f(⇤+ i)=f(⇤) 2 2. If ⇤ and ⇤ are such that ⇤ ⇤ and ⇤ ⇤ = then 1 2 1 [ 2 2S 1 \ 2 ; f(⇤ )+f(⇤ ) f(⇤ ⇤ ) 1 1 1 [ 2 60 We can now prove the following theorem h Proposition 41. S↵ (µ) exist and we have h 1 S↵ (µ)= inf ⇤ Eµ [p↵,⇤](3.2.1) ⇤ ⇤ 2 | | Proof. Our proof relies on the Lemma ??. According to this Lemma it is su cient to prove that Eµ [p↵,⇤]issubadditiveforsets⇤1 and ⇤2 that ⇤1 ⇤2 and [ 2S 1 ⇤ ⇤ = .ForanytilingT ⌦, if we define T to be the word h (a, 0)h (b, 0) 1 \ 2 ; 2 [a,b] T T in Zm Zn, we have for every a p (T ) p (T )+p (T )(3.2.2) ↵,[a,c] ↵,[a,b] ↵,[b,c] Indeed Let [a,c] be a subword of T[a,c] such that L [a,c] = p↵,[a,c] (T )thenif [a,c] is also a subword of T[a,b] or T[b,c] we have p (T ) max p (T ) ,p (T ) p (T )+p (T )(3.2.3) ↵,[a,c] ↵,[a,b] ↵,[b,c] ↵,[a,b] ↵,[b,c] Otherwise there exist two words [a,b] and [b,c] such that [a,b] [b,c] = [a,c] (3.2.4) where the product stands for the concatenation in Zm Zn and [a,b] (resp. [b,c])is ⇤ asubwordofT (resp. T ). But then this imposes l(T ) p (T )(resp. [a,b] [b,c] [a,b] ↵,[a,b] l(T ) p (T )). We have no shown that the function is subbaditve and since [b,c] ↵,[b,c] the translation invariance of Eµ [p↵,⇤]comesstraightfromthetranslationinvariance v of µ, this finishes our proof. We can do the exact same proof for the existence of S↵. 61 As expected for our construction the function S↵ characterize the limiting be- havior of a measure and we have the following theorem Proposition 42. Let µ be an ergodic Gibbs measure, there is at most one word ! such that S! (µ) > 0. Proof. This is a straight consequence from the last section since we proved that any ergodic Gibbs measure is supported on at most one geodesic. This also tells us that for any two word ↵1 and ↵2 in group we have: h v S↵1 S↵2 > 0 if and only if ↵1 = ↵2. We are now able to define the surface tension relative to a periodic geodesic ↵Z. Let us consider the essential ↵ slope S¯ of a configuration T ⌦defineby ↵ 2 1 S¯↵ (T ) = lim inf ⇤ p↵,⇤ (T ) . ⇤ | | The existence of the essential slope is a direct consequence of Proposition 37. ¯ Definition 43. For every finite word ↵ in Zm Zn,denote ↵ the closure for the ⇤ S topology of local convergence of the subset µ (⌦, ):S¯ (µ) > 0 .Foranyu 2P F⌧ ↵ in (0, 1)2 we define he ↵-surface tension ↵ (u)= inf SFE (u) . µ ¯:S↵(µ)=u { 2S } where SFE is the entropy per unit of the measure. 62 For a detailed definition and construction of the SFE the reader can look in appendix A. 3.2.2 Minimizers of the entropy exist and are Gibbs mea- sures In this section we will prove that this infimum is in fact a minimum and that this minimum is always reached by a Gibbs measure. This is the first half of a variational principle. Proposition 44. For every finite word ↵,If ↵ (u) at u then there exist a measure µ(↵, u) in S¯↵ with slope u such that SFE (µ(↵, u)) = ↵ (u) Proof. Since the infimum of upper semicontinuous functions is still upper semicon- tinuous, if we can prove that µ Eµ [p↵,⇤]isuppersemicontinuouswewillobtain ! at the same time that S↵ is also upper semi continuous. . But this is straight- forward because Eµ [p↵,⇤]dependscontinuouslyontheprobabilitiestoreadtheword !⇤ associated to the tiling configuration of any box containing ⇤. The upper continuity of S tells us that the level sets M = µ : S (µ) c ↵ ↵,c { ↵ } are closed for the topology of local convergence. Moreover we also have that for 2 any m Z with positive coordinates the level sets µ : m, S↵ (µ) c are closed 2 { h i } since the sum of a finite number of upper semicontinuous function is also upper semicontiuous. Finally the level sets µ : SFE (µ)+ m, S (µ) c are compact { h ↵ i } because they are closed sets included in the compact set µ : SFE (µ) c .This { } 63 tells us that inf µ : SFE (µ)+ m, S↵ (µ) c µ (⌦,F ) { h i } 2P is a minimum for any m with positive coordinates. For every u such that ↵ (u)is strictly convex at u,wecanchoosem(u)suchthatthismiminumisreachedinu which finishes the proof. In particular the specific free energy reaches a global minimum and we have the following corollary Corollary 45. For every finite word ↵, there exist a measure µ(↵) in S¯↵ such that SFE (µ(↵)) = inf ↵ (u) u The following theorem tells us that if a measure reaches this minimum then it is necessarily a Gibbs measure which is the first half of a variational principle Theorem 46. Let µ be a translation invariant measure such that with S↵(µ)=u and such that u>0 and SFE(u)= ↵(u). Then µ is a Gibbs measure Proof. Suppose that µ is such that S↵(µ)=u and µ is not a Gibbs measure. We will show that it is always possible to modify µ to obtain a translation invariant measure ⌫ such that S↵(⌫) If µ is not a Gibbs measure then there exist a finite set ⇤such that Eµ [ ⇤c ] = ⇤c ( ) , ⇧|F 6 ⇧ 64 Now let M be the diameter of ⇤, we consider the modified measure ⌫ = µ ⇤+(kmM,lnM). Yk,l where ⇤+(kM,lM) is the kernel associated to the uniform measure on tilings of ⇤+ (kM, lM). By construction we know that for any (k, l) =(k0,l0)thetwosets⇤+ 6 (kmM, lnM)and⇤+(k0mM, l0nM) are disjoints. While ⌫ is not necessarily ⌧ invariant, the average 1 ⌫¯ = ⌫ ⌧ (i, j) M 2 i,j M X is invariant and we have SFE(¯⌫)=SFE(⌫). Finally we must have SFE(⌫) < SFE(µ) because by assumption we know that E⌫ [ ⇤c ] < Eµ [ ⇤c ] ⇧|F ⇧|F And the kernel is applied on a positive portion of the plane. This concludes our proof 3.3 Construction of invariant measures with a given slope In this section there will be no mathematical results but we will describe qualitatively how to simulate ergodic Gibbs measures. We give our self a periodic geodesic !Z and a rational slope p/q.Theideaisto use ”cannonical” tilings of tori with extremal slopes and supported on !Z. and try to combine them in order to reach the desired slope. 65 In the following example we choose the geodesic !Z =(abab2abababa2bab2abababa)Z 10 5 with an horizontal speed sh = 9 and a vertical speed sv = 9 . We begin by constructing a torus with zero vertical speed. In order do that we 2 look at the only path p in Z such that projgroup(p)=! and such that the departure and final points of p gave same abscissa. This can be done by an induction on the i n i subwords of ! where at each step we chose b or b depending on which one has the closest abcyssa to the starting point. It is import an to notice that the same technique can be applied for any direction int the plane. In our case we obtain 1 2 2 2 p = abab ababab a bab abababa. We can now insert bands of horizontal tilings. The e↵ect of this adding is to decrease the slope of our configuration since the height di↵erence across one band is flat. In the end we obtain a configuration as shown in Figure 3.3.1. We now do the same to construct a tiling with extremal slope, In the sense that the vertical slope is the same as the horizontal slope. It is interesting to notice that the tiling obtained in one of the extremal tiling described in Chapter 2. By combining the two configurations of the two tori one can obtain a bigger tori supported on the same geodesic and with the wanted slope as shown in Figure 3.3.2. 66 Figure 3.3.1: Torus with null vertical slope Figure 3.3.2: Torus with Extremal slope We can now use the Glauber dynamics described in Chapter 2 to randomize the configuration. Theorem 2 assure us that we can obtain any given configuration with same horizontal and vertical words. The result look like Figure 3.3.4 To obtain the image of an ergodic measure we need to check that the white lines are evenly spread in the picture. This is a consequence of the ergodic theorem. The other possibility would be a picture where the lines are interacting with each other and form a cluster. The resulting measure would be the convex combination of a flat ergodic measure ad an ergodic measure with extremal slope. All the simulations we have made show that the white lines never gather independently of the slope or the geodesic chosen. This would means that we can construct any ergodic measure with 67 Figure 3.3.3: Combined configuration agivenslopeandperiodicgeodesicwiththismethod. Unfortunately we have not been able to prove this result in the general, this would require to prove that the variance of the height function distance between to fixed points is linear in the distance. However in the case where the vertical speed (resp. horizontal speed) is null and the word omega only contains brick of width two, as shown in the bottom left band of the last picture, one can construct an injection from n merged lines to n non interacting lines. This is shown by the red line in the picture which is used to split one line from the others and is represented as a separated line in a band in the image below. In particular this means that the random variable encoding the number of lines being crossed between two fixed points will have a log-concave law. one can show that this implies that the lines are actually evenly spread on the torus and the 68 Figure 3.3.4: Combined configuration after shu✏ing resulting measure is ergodic. 69 Chapter Four Conclusion 71 When then this the is was written our goal was to understand which results and limiting behaviors of domino tilings could be generalized to some well chosen non- integrable tilings. In this conclusion we will try to expose some the open problems that seem the most relevant to me and what could lead to their proof. 4.1 Existence and characterization of the ergodic Gibbs measures One of the conclusions of Theorem 39 in Chapter 3 is that, once we know which set of geodesics support an ergodic measure, the pre-image of the projection on this geodesic should look like the level-sets of a semi-norm. On a macroscopic scale those pictures can be characterized by the asymptotic direction of the level sets and their density, or equivalently, by the average number of level set crossed in the vertical and horizontal directions. In terms of height function, this corresponds to the limiting speeds at which the height function travels in the graph in those directions. This behavior is what we observed in the picture 1.2.1. It is thus natural to attempt to construct ergodic Gibbs measures by fixing a periodic geodesic and a couple of asymptotic vertical and horizontal speed. And the first most natural result would be to prove the existence of such measures. Conjecture 47. Let ! be a finite word in Zm Zn. Then for any slope s = ⇤ (s ,s )such that s The most challenging part of the proof of this conjecture would be to show that the limiting measure, when expanding the size of the tori, is a single ergodic Gibbs 72 measure and not a mixture of those. We expect to prove this by showing that Gibbs modifications of a translation invariant tiling of Z2 can only converge to a limiting measure whose ergodic components must all have the same speed. Indeed, we have the following result: Lemma 48. Let µ be a translation invariant measure with ergodic components of di↵erent speed, then there exists ✏>0 such that 2 Var [dZm Zn (0,x)] ✏ x , ⇤ | | Thus the crucial step would be to prove that the following conjecture is true. Conjecture 49. There exists a constant C such that, for any finite region R, we have VarR [dZm Zn (0,x)] C x ⇤ | | In my opinion Conjecture 49 could be proven by using Doub’s martingales and martingale concentrations inequalities. The main point would be to show that a small modification of the boundary conditions does not change the expectation of the distance in the graph between two fixed points by more than a constant. Proving this conjecture would be an essential result in the understanding of Gibbs measure on tilings in general, and is the problem we are currently focus on. Until now we assumed that the ergodic Gibbs measure were supported on a single periodic geodesic of the group Zm Zn because it is the easiest and most natural way ⇤ to obtain a translation invariant measure. However, this might be too restrictive. Indeed, if we look at a geodesic as an infinite word in the alphabet for which the letters are the di↵erent types of tiles modulo translations of length 3, we have the 73 following result: Lemma 50. Let be a subshift in the alphabet given by the di↵erent type of tiles and S and be two elements of . Then if we denote by p and p the projections 1 2 S 1,n 2,n on 1 and 2 truncated at n, we have that for any double infinite word ↵: lim p ,n (↵)= lim p ,n (↵) . n 1 n 2 !1 !1 This suggests the following conjecture: Conjecture 51. For any ergodic shift and any couple of speed (s ,s ) such that S h v s + s 1, there exists an ergodic measure supported on . | h| | v| S 4.2 Variational principle and unicity of ergodics Gibbs measures When looking at ergodic Gibbs measures, it is natural to expect that they can be characterized as the minimizers of the entropy amongst measures with a given surface tension. In the domino case, the surface tension is characterized by the slope of the measure and She eld [26] showed that for a fixed slope, the ergodic ergodic Gibbs measures are exactly the minimizers of the entropy. For tiling by bars we can still characterize the surface tension according to the limiting speeds described in chapter 3, but we must additionally specify precisely in which direction the height function is travelling in the graph, that is specify precisely which geodesic actually supports our measure. We have already proven in chapter 3 that for a given a slope and geodesic the minimizers of the specific free energy are Gibbs measure but the converse result must be true. 74 The key step in order to prove this results is to show that ergodic measures can somehow approximate any given boundary condition with roughly the same slope. The corresponding property would be that an annular finite region whose boundaries heights functions are not “far” in the Cayley graph of Zm Zn can actually be tiled. ⇤ But this is exactly the result of Theorem 6. Thus believe that the variational principle still holds for tiling by bars. Another very important result would be to prove the unicity of ergodic Gibbs measure when all the parameters are fixed. Conjecture 52. Let (s ,s ) be a couple of slopes such that s + s 1 and ! be h v | h| | v| a finite word in Zm Zn. There exists a unique ergodic Gibbs measure supported on ⇤ ! with slopes (sh,sv) . AverystrongimplicationofProposition1isthatthetilingpicturesall“look the same” in the sense that when looking at the pre-image of the projection of the height function on !,oneobtainsasetofrandompathcrossingtheplaneinthe same direction as the zero set for our semi-norm. Now, if we look at the product measure obtained when superimposing two ergodic pictures of the same slope and supported on the same geodesic, we still have a lot of informations on the resulting images in terms of those paths. In some way, all the macroscopic characteristics of the picture described in [26] are still valid. The main obstacle to complete the proof would be to define an analogous of the cluster swap. This should be a local move which allows us to switch paths when they cross, but does not decrease the entropy of the product measure. We believe this should be doable at least for some well chosen geodesics. Arnother interesting problem would be to find explicit bounds for the distance from the height function to the geodesic on which it is supported. We believe that, 75 using the same type of technique found in [3], one could prove that the probability to be far from the geodesic must decrease exponentially. Those results seems do not seem to be strongly dependent on the tiling model. The last question we will mention concerns the triviality of the maximal projection group of the Conway group.This is closely related to the existence of a surface tension. It seems to me that, if this group is not trivial (or finite), then one can create a surface tension from boundary conditions. I have no clear answer about the converse result, but it would be great to prove that, when the tiling group is trivial, this cannot happen. Problem 53. Is the non-triviality of the the natural projection group of the Con- way group directly related to surface tensions and to “arctic circle” types limiting behaviors? In the case of tilings by vertical dominos and one type of corner dominos, one can show that the group is trivial and that there can be at most one ergodic Gibbs measure, which support this conjecture. An interesting model to study would be 3 1 rectangles together with corner tiles. Ib this model the height function is in ⇥ Z/3Z and It is natural to conjecture that there should be a finite number of extremal Gibbs measures Appendix A Construction of the specific free energy 77 In this appendix we describe how to construct the specific free energy of a measure µ on tilings. In physics specific free energy is often referred as entropy per site, it can been seen as the rate at which the number of configuration grows when we consider a sequence of increasing region. Most of the proofs in this appendix are adaptation of the one find in []georgii A.1 Entropy of a mesure Throughout this section we will consider (⌦, )tobeanarbitrarymeasurablespace F and µ and ⌫ to be tow probability measures on (⌦, ). F Definition 54. For any sigma sub algebra of we define A F (µ ⌫)=⌫ (f log f ) HA | A A if µ ⌫ and (µ ⌫)= otherwise and where f is the Radon-Nikodym derivative ⌧ HA | 1 A of µ relative ot ⌫ |A |A The entropy retaltive to a sigma algebra has the following properties Proposition 55. For any sigma sub algebra and any measures µ and ⌫ we have A 1. (µ ⌫) is an increasing function of HA | A 2. (µ ⌫) 0 HA | 3. (µ ⌫)=0if and only if µ = ⌫ on HA | A 4. (µ ⌫)=0is a convex function of the pair (µ, ⌫) HA | 78 Proof. If we consider two sigma algebra 1 and 2 such that 1 2.Letf 1 anf A A A ⇢A A f 2 be the Radon Nikodym derivatives of µ relatives to ⌫.Ifwedenoteby the A function definded by (x)=1 x + x log x we can write the equality (µ ⌫)=⌫ ( f ) (A.1.1) HA | A Now the convexity of and Jensen sinequalitygivesusthat; \ 1 (µ ⌫)=⌫ ( ⌫ (f 2 1)) HA | A |A = ⌫ (⌫ ( f 2 1)) A |A 2 (µ ⌫) HA | which gives us the first point. The second point follows directly from the first one and the observation that ,⌦ (µ ⌫)=0. H{; } | For the third point, it is a straight consequence of the equality (A.1.1) since (µ ⌫)=0implies f =0⌫-a.s and thus f =1⌫-a.s so that µ = ⌫. HA | A A Finally if we consider four measures µ1,µ2,⌫1,⌫2 and write for 0 < <1,µ ¯ = µ +(1 )µ and bar⌫ = ⌫ +(1 )⌫ .IfwedenotetherespectinRadon- 1 2 1 2 i i Nykodym derivatives µi = f µ¯ and ⌫i = g ⌫¯,theconvexityof gives us A A (¯µ ⌫¯)=¯⌫ sg1 f 1 +(1 s)g2 f 2 HA | A A A A ⌫¯ sg 1 (f 1 )+(1 s)g2 ( f 2 ) A A A A s (µ1 ⌫1)+(1 s) (µ2 ⌫2) HA | HA | 79 which conclude the proof of the last property. We will now prove can be defined independently of sigma algebras as a sup of all relative entropy. Proposition 56. Let µ and ⌫ be two probability measures on (⌦, ) and ( ↵)↵ be F A 2I an increasing family of sigma subalgebras of . If we denote = ↵ ↵ the F A 2I A smallest sigma algebra containing all ’s. We have S A↵ (µ ⌫)=lim ↵ (µ ⌫)=sup ↵ (µ ⌫) HA | ↵ HA | ↵ HA | 2I 2I Proof. From the first point Proposition 55 we already have the following inequalities (µ ⌫) lim ↵ (µ ⌫)=sup ↵ (µ ⌫) HA | ↵ HA | ↵ HA | 2I 2I Hence we just need to prove that (µ ⌫) sup↵ ↵ (µ ⌫). If sup↵ ↵ (µ ⌫)= HA | 2I HA | 2I HA | then the result is trivial. If not there exist a family of well defined function (f↵)↵ 1 2I such that µ = f ⌫ on Moreover this is a well know property of conditional ex- ↵ A↵ pectations that (f↵)↵ is a martingale relative to ⌫.Letc =sup↵ ↵ (µ ⌫). For 2I 2I HA | r 0, We can write 1 c +1 ⌫ f↵1 f↵ r (log r) ⌫ f↵ log f↵1 f↵ r { } { } log r for all ↵,sothatthe(f↵)↵ are uniformly ⌫-integrable. It is also a Cauchy net 2I since any uniformly integrable martingale is convergent in L1(⌫). The completeness 1 of L (⌫)tellsusthat(f↵)↵ converges to a single limit f in (f↵)↵ and that 2I 2I we must necessarily have µ = g⌫ on . Finally by considering the functions gN = A N N max (f log f,N)andg↵ =max(f↵ log f↵,N), we see that g↵ converges in probability 80 to gN .Thus N N ⌫(g )=lim⌫ g↵ lim ↵ (µ ⌫)=c ↵ ↵ A 2I 2I H | When N tends to infinity this concludes our proof If we choose to be the sigma algebra this allows us to give a general meaning A F to the notion of entropy. A.2 Construction of the specific free energy We could recall the definition of entropy in a more general settings but since we are looking at discrete systems it is su cient for us to use the Shannon Entropy. In all this appendix we consider that the set of prototiles is fixed and we denote by G the Conway group of this prototiles. We also suppose that the height function unequaly characterize any tilings as i the case for tilings by bars. Definition 57. We fix now (⌦, )tobethespaceoftilingstogetherwiththesigma F algebra generated by cylinder events as discibed in chapter 2. Let ⇤be a finite region 2 of Z .Theentropy Lambda(µ)ofameasureµ on ⇤is H| (µ)= µ (T ⇤c)logµ (T ⇤c) H⇤ | | T G⇤ X2 Our goal will be to show that for any increasing sequence of regions (⇤n)n 1 such 2 that n 1 ⇤n = Z ,thequantity S 1 SFE (µ)= lim ⇤n ⇤ (µ) n n !1 | | H 81 is well defined and independent of (⇤n)n 1. We begin by proving the supperaditivity of the entropy Proposition 58. Let ⇤ and be two finite region of the plan, we have the following inequality: ⇤ (µ)+ (µ) ⇤ (µ)+ ⇤ (µ) H H H \ H [ Proof. Let fR be the density of µ for the uniform measure R on tilings of a region R with the convention that f =1.Ifweset⌫ = µ ⇤,wecanconstructthe ; \ f ⇤ f functions [ and which are defined ⌫-a.s. Now if we apply the third property f⇤ f ⇤ \ of proposition 55 we obtain f ⇤ ⇤ (µ) ⇤ (µ)=µ log [ H H [ f ✓ ⇤ ◆ = ⇤ (µ ⌫) H [ | (µ ⌫) H | f µ log f ⇤ ✓ \ ◆ ⇤ (µ) (µ) H\ H which completes our proof This tells us that when the measure µ is ⌧-translation invariant then the func- tion :⇤ is also supperaditive and ⌧-translation invariant. We have now !Hµ everything to prove the main theorem of this appendix Proposition 59. Let µ be a ⌧-translation invariant measure on (⌦, ). For every F sequence of cubes (⇤n)n 1 such that ⇤n , the limit | |!1 1 SFE (µ)= lim ⇤n ⇤ (µ) n n !1 | | H 82 exists and we have 1 SFE (µ)=inf ⇤ ⇤(µ) ⇤ 2C | | H where is the set of rectangles in Z2. C Proof. The proof is very similar to the proof of the subbaditive theorem we used in 1 chapter 2. Let c =inf⇤ ⇤ ⇤(µ)and⇤✏ be a rectangle such that 2C | | H 1 ⇤ (⇤ ) c ✏. | ✏| ✏ If (⇤n)n 1 is a sequence of cube with ⇤n ,Wecanwriteeach⇤n as a disjoint | |!1 union of Nn translates of ⇤✏ together with a set with area bounded by a constant M.Thesubbaditivityofthefunction gives us that (⇤ ) N (⇤ )+ ((0, 0)) n n ✏ | | M When n ,ifwedividethisinequalityby ⇤n ,wehave 0andtheinequality !1 | | Nn ! becomes 1 1 lim sup ⇤n (⇤n)=limsupNn ⇤✏ (⇤n) n | | n | | !1 !1 1 ⇤ (⇤ ) c + ✏ |✏| n Which is exactly our result. Appendix B Notations 84 Tilings and height functions Z2 The square lattice in the plane Fa,b Free group generated by a and b CG Cayley graph of G t ,...,t set of prototiles { 1 k} R¯ Boundary of the region R m n Zm Zn Group with presentation a, b a = b = e ⇤ h | i dG Distance associated to the group G h(x) Heightfunctionatvertex x @(G) Boundary of the group G !g, Geodesic form the point g to 1 1 S!g, Extremal set associated to !g, 1 1 S! Extremal set associated to the geodesic word ! 1 1 projG Projection on the group G hT Height function associated to the tiling T Txi,gi, Extremal tiling with height g at the vertex x and going in the direction 1 , 1 l(g)Distancefromg to the origin in the Cayley graph R(T1,T2)DistancebetweenthetilingT1 and T2 T !h Path word described by a torus tiling T following the paths from the (0, 0) to (0,n) T !v Path Word described by a torus tiling T following the paths from the (0, 0) to (n, o) ↵, hT Height function of the tiling T for path which ends in the ↵, torus Translation invariant Gibbs measures on tilings 2 ⌦SetoffunctionfromZ to Zm Zn ⇤ 85 Borel sigma algebra of the product topology on ⌦ F Sigma-algebra with respect to which h(x)ismeasurableforallx S FS 2 ⇤Rc (.) Uniformmeasureontilingsof R conditioned on !Rc Set of Gibbs measure on tilings of the plane G Tail sigma-algebra I ⌧ Group of translations generated by x x +(0,m)andx x +(n, o) ! ! ⌧ sigma algebra generated by translation invariant events F (⌦, ⌧ )Setoftranslationinvariantmeasureson⌦ P F µ measure µ ( A) A ·| Set of translation invariant Gibbs measures G⌧ ↵ ( ,x)Cocycleactingthrough at x DT (x, y)distancedZm Zn (hT (x),hT (y)) ⇤ 2 s Limiting speed associated to the vector in R (T, )BoundarypointassociatedtothetilingT in the direction p (!)Projectiondefinedbymaxl ( ): is a subword of both ↵ and ↵ ↵0,[a,b] { 0} S↵ (µ)Speedofµ relative to the geodesic of period ↵ SpaceofintervalinZ ↵ Surface tension relative to the geodesic of period ↵ Bibliography [1] Dani`ele Beauquier, Maurice Nivat, Eric R´emila, and Mike Robson. Tiling figures of the plane with two bars. Computational Geometry,5(1):1–25,1995. [2] J-R Chazottes, J-M Gambaudo, and Fran¸cois Gautero. 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