Lectures on random lozenge tilings Vadim Gorin University of Wisconsin–Madison, Massachusetts Institute of Technology, and Institute for Information Transmission Problems of Russian Academy of Sciences. [email protected] This material will be published by Cambridge University Press as “Lectures on Random Lozenge Tilings” by Vadim Gorin. This prepublication version is free to view and download for personal use only. Not for redistribution, resale or use in derivative works. Contents About this book page 6 1 Lecture 1: Introduction and tileability. 7 1.1 Preface 7 1.2 Motivation 9 1.3 Mathematical questions 10 1.4 Thurston’s theorem on tileability 15 1.5 Other classes of tilings and reviews. 20 2 Lecture 2: Counting tilings through determinants. 21 2.1 Approach 1: Kasteleyn formula 21 2.2 Approach 2: Lindstrom-Gessel-Viennot¨ lemma 25 2.3 Other exact enumeration results 30 3 Lecture 3: Extensions of the Kasteleyn theorem. 31 3.1 Weighted counting 31 3.2 Tileable holes and correlation functions 33 3.3 Tilings on a torus 34 4 Lecture 4: Counting tilings on large torus. 39 4.1 Free energy 39 4.2 Densities of three types of lozenges 41 4.3 Asymptotics of correlation functions 44 5 Lecture 5: Monotonicity and concentration for tilings 46 5.1 Monotonicity 46 5.2 Concentration 48 5.3 Limit shape 50 6 Lecture 6: Slope and free energy. 52 6.1 Slope in a random weighted tiling 52 6.2 Number of tilings of a fixed slope 54 6.3 Concentration of the slope 56 6.4 Limit shape of a torus 57 3 4 Contents 7 Lecture 7: Maximizers in the variational principle 58 7.1 Review 58 7.2 The definition of surface tension and class of functions 59 7.3 Upper semicontinuity 62 7.4 Existence of the maximizer 64 7.5 Uniqueness of the maximizer 65 8 Lecture 8: Proof of the variational principle. 67 9 Lecture 9: Euler-Lagrange and Burgers equations. 74 9.1 Euler-Lagrange equations 74 9.2 Complex Burgers equation via a change of coordinates 75 9.3 Generalization to qVolume–weighted tilings 78 9.4 Complex characteristics method 79 10 Lecture 10: Explicit formulas for limit shapes 81 10.1 Analytic solutions to the Burgers equation 81 10.2 Algebraic solutions 84 10.3 Limit shapes via quantized Free Probability 86 11 Lecture 11: Global Gaussian fluctuations for the heights. 90 11.1 Kenyon-Okounkov conjecture 90 11.2 Gaussian Free Field 92 11.3 Gaussian Free Field in Complex Structures 96 12 Lecture 12: Heuristics for the Kenyon-Okounkov conjecture 98 13 Lecture 13: Ergodic translation-invariant Gibbs measures 105 13.1 Tilings of the plane 105 13.2 Properties of the local limits 107 13.3 Slope of EGTI measure 109 13.4 Correlation functions of EGTI measures 111 13.5 Frozen, liquid, and gas phases 112 14 Lecture 14: Inverse Kasteleyn matrix for trapezoids. 116 15 Lecture 15: Steepest descent method for asymptotic analysis. 123 15.1 Setting for steepest descent 123 15.2 Warm up example: real integral 123 15.3 One-dimensional contour integrals 124 15.4 Steepest descent for a double contour integral 126 16 Lecture 16: Bulk local limits for tilings of hexagons 129 17 Lecture 17: Bulk local limits near straight boundaries 138 5 18 Lecture 18: Edge limits of tilings of hexagons 145 18.1 Heuristic derivation of two scaling exponents 145 18.2 Edge limit of random tilings of hexagons 147 18.3 Airy line ensemble in tilings and beyond 152 19 Lecture 19: Airy line ensemble, GUE-corners, and others 154 19.1 Invariant description of the Airy line ensemble 154 19.2 Local limits at special points of the frozen boundary 156 19.3 From tilings to random matrices 157 20 Lecture 20: GUE–corners process and its discrete analogues. 164 20.1 Density of GUE–corners process 164 20.2 GUE–corners process as a universal limit 168 20.3 A link to asymptotic representation theory and analysis 171 21 Lecture 21: Discrete log-gases. 176 21.1 Log-gases and loop equations 176 21.2 Law of Large Numbers through loop equations 179 21.3 Gaussian fluctuations through loop equations 182 21.4 Orthogonal polynomial ensembles 185 22 Lecture 22: Plane partitions and Schur functions. 188 22.1 Plane partitions 188 22.2 Schur Functions 190 22.3 Expectations of observables 192 23 Lecture 23: Limit shape and fluctuations for plane partitions. 199 23.1 Law of Large Numbers 199 23.2 Central Limit Theorem 205 24 Lecture 24: Discrete Gaussian component in fluctuations. 209 24.1 Random heights of holes 209 24.2 Discrete fluctuations of heights through GFF heuristics. 210 24.3 Approach through log-gases 214 24.4 2d Dirichlet energy and 1d logarithmic energy. 217 24.5 Discrete component in tilings on Riemann surfaces 224 25 Lecture 25: Sampling random tilings. 225 25.1 Markov Chain Monte-Carlo 225 25.2 Coupling from the past [ProppWilson96] 229 25.3 Sampling through counting 233 25.4 Sampling through bijections 233 25.5 Sampling through transformations of domains 234 References 236 Index 248 About this book These are lecture notes for a one semester class devoted to the study of random tilings. It was 18.177 taught at Massachusetts Institute of Technology during Spring of 2019. The brilliant students who participated in the class1: Andrew Ahn, Ganesh Ajjanagadde, Livingston Albritten, Morris (Jie Jun) Ang, Aaron Berger, Evan Chen, Cesar Cuenca, Yuzhou Gu, Kaarel Haenni, Sergei Ko- rotkikh, Roger Van Peski, Mehtaab Sawhney, Mihir Singhal provided tremen- dous help in typing the notes. Additional material was added to most of the lectures after the class was over. Hence, when using this review as a textbook for a class, one should not expect to cover all the material in one semester, something should be left out. I also would like to thank Amol Aggarwal, Alexei Borodin, Christian Krat- tenthaler, Igor Pak, Jiaming Xu, Marianna Russkikh, and Semen Shlosman for their useful comments and suggestions. I am grateful to Christophe Charlier, Maurice Duits, Sevak Mkrtchyan, and Leonid Petrov for the help with simula- tions of random tilings. Funding acknowledgements. The work of V.G. was partially supported by NSF Grants DMS-1664619, DMS-1949820, by NEC Corporation Fund for Research in Computers and Communications, and by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison with funding from the Wisconsin Alumni Research Foun- dation. Chapters 6 and 13 of this work were supported by Russian Science Foundation (project 20-41-09009). 1 Last name alphabetic order. 6 Lecture 1: Introduction and tileability. 1.1 Preface The goal of the lectures is to understand the mathematics of tilings. The general setup is to take a lattice domain and tile it with elementary blocks. For the most part, we study the special case of tiling a polygonal domain on the triangular grid (of mesh size 1) by three kinds of rhombi that we call “lozenges”. The left panel of Figure 1.1 shows an example of a polygonal domain on the triangular grid. The right panel of Figure 1.1 shows the lozenges: each of them is obtained by gluing two adjacent lattice triangles. A triangle of the grid is surrounded by three other triangles, attaching one of them we get one of the three types of lozenges. The lozenges can be also viewed as orthogonal projections onto the x+y+z = 0 plane of three sides of a unit cube. Figure 1.2 provides an example of a lozenge tiling of the domain of Figure 1.1. Figure 1.3 shows a lozenge tiling of a large domain, with the three types of lozenges shown in three different colors. The tiling here is generated uniformly at random over the space of all possible tilings of this domain. More precisely, it is generated by a computer that is assumed to have access to perfectly ran- dom bits. It is certainly not clear at this stage how such “perfect sampling” may be done computationally, in fact we address this issue in the very last lecture. Figure 1.3 is meant to capture a “typical tiling”, making sense of what this means is another topic that will be covered in this book. The simulation re- veals interesting features: one clearly sees next to the boundaries of the domain formation of the regions, where only one type of lozenges is observed. These regions are typically referred to as “frozen regions” and their boundaries are “artic curves”; their discovery and study has been one of the important driving forces for investigations of the properties of random tilings. We often identify a tiling with a so-called “height function”. The idea is to think of a 2-dimensional stepped surface living in 3-dimensional space and treat tiling as a projection of such surface onto x + y + z = 0 plane along the 7 8 Lecture 1: Introduction and tileability. B = 5 C = 5 A = 5 Figure 1.1 Left panel: A 5×5×5 hexagon with 2×2 rhombic hole. Right panel: three types of lozenges obtained by gluing two adjacent triangles of the grid. B = 5 C = 5 A = 5 Figure 1.2 A lozenge tiling of a 5 × 5 × 5 hexagon with a hole. (1;1;1) direction. In this way three lozenges become projections of three el- ementary squares in three-dimensional space parallel to each of the three co- ordinate planes. We formally define the height function later in this lecture.