Tiling by Bars

Tiling by bars by Martin Tassy B.Sc., ENS Cachan 2007 M.A., UniversitéPierre 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 Lycée de Provence, he entered classes préparatoires successively at Lyce Thiers and Lycée Sainte-Genevieve. In 2006, he was admitted at Ecole´ normal supérieure de Cachan. During his four years as a étudiant normalien, he received a Bachelor of Science in Mathematics, a Master of Mathematics in Probabilities from UniversitéRennes 1, a Master in Financial Mathematics from UniversitéParis 6 in 2008, and the french Aggrégation 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 Sheffield 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 connectivity 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 measures 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.

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