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El. knyga: Lectures on Random Lozenge Tilings

(University of Wisconsin, Madison)
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"Over the past 25 years, there has been an explosion of interest in the area of random tilings. The first book devoted to the topic, this timely text describes the mathematical theory of tilings. It starts from the most basic questions (which planar domains are tileable?) before discussing advanced topics about the local structure of very large random tessellations. The author explains each feature of random tilings of large domains, discussing several different points of view and leading on to open problems in the field. The book is based on upper-division courses taught to a variety of students, but it also serves as a self-contained introduction to the subject. Test your understanding with the exercises provided, and discover connections to a wide variety of research areas in mathematics, theoretical physics, and computer science, such as conformal invariance, determinantal point processes, Gibbs measures, high-dimensional random sampling, symmetric functions, and variational problems"--

Recenzijos

'The lectures provide connections between random tilings and many areas in mathematics and theoretical physics, and include an exhaustive reference list. Mathematicians and others interested in the hows and whys of this intriguing area would be well-served by consulting this text.' Thomas Polaski, Mathematical Reviews/MathSciNet 'It seems that the reviewed book is the first introductory text about this fascinating topic. The release of this book is a great event for everyone interested in this problem.' Anton Shutov, zbMATH

Daugiau informacijos

This is the first book dedicated to reviewing the mathematics of random tilings of large domains on the plane.
Preface ix
1 Lecture 1: Introduction and Tileability
1(15)
1.1 Preamble
1(2)
1.2 Motivation
3(1)
1.3 Mathematical Questions
4(5)
1.4 Thurston's Theorem on Tileability
9(5)
1.5 Other Classes of Tilings and Reviews
14(2)
2 Lecture 2: Counting Tilings through Determinants
16(11)
2.1 Approach 1: Kasteleyn Formula
16(4)
2.2 Approach 2: Lindstrom-Gessel-Viennot Lemma
20(5)
2.3 Other Exact Enumeration Results
25(2)
3 Lecture 3: Extensions of the Kasteleyn Theorem
27(8)
3.1 Weighted Counting
27(2)
3.2 Tileable Holes and Correlation Functions
29(1)
3.3 Tilings on a Torus
30(5)
4 Lecture 4: Counting Tilings on a Large Torus
35(7)
4.1 Free Energy
35(2)
4.2 Densities of Three Types of Lozenges
37(3)
4.3 Asymptotics of Correlation Functions
40(2)
5 Lecture 5: Monotonicity and Concentration for Tilings
42(6)
5.1 Monotonicity
42(2)
5.2 Concentration
44(2)
5.3 Limit Shape
46(2)
6 Lecture 6: Slope and Free Energy
48(6)
6.1 Slope in a Random Weighted Tiling
48(2)
6.2 Number of Tilings of a Fixed Slope
50(2)
6.3 Concentration of the Slope
52(1)
6.4 Limit Shape of a Torus
53(1)
7 Lecture 7: Maximizers in the Variational Principle
54(9)
7.1 Review
54(1)
7.2 The Definition of Surface Tension and Class of Functions
55(2)
7.3 Upper Semicontinuity
57(3)
7.4 Existence of the Maximizer
60(1)
7.5 Uniqueness of the Maximizer
61(2)
8 Lecture 8: Proof of the Variational Principle
63(7)
9 Lecture 9: Euler-Lagrange and Burgers Equations
70(7)
9.1 Euler-Lagrange Equations
70(1)
9.2 Complex Burgers Equation via a Change of Coordinates
71(3)
9.3 Generalization to Volume-Weighted Tilings
74(1)
9.4 Complex Characteristics Method
75(2)
10 Lecture 10: Explicit Formulas for Limit Shapes
77(9)
10.1 Analytic Solutions to the Burgers Equation
77(4)
10.2 Algebraic Solutions
81(1)
10.3 Limit Shapes via Quantized Free Probability
82(4)
11 Lecture 11: Global Gaussian Fluctuations for the Heights
86(8)
11.1 Kenyon-Okounkov Conjecture
86(2)
11.2 Gaussian Free Field
88(4)
11.3 Gaussian Free Field in Complex Structures
92(2)
12 Lecture 12: Heuristics for the Kenyon-Okounkov Conjecture
94(8)
13 Lecture 13: Ergodic Gibbs Translation-Invariant Measures
102(11)
13.1 Tilings of the Plane
102(2)
13.2 Properties of the Local Limits
104(2)
13.3 Slope of EGTI Measure
106(2)
13.4 Correlation Functions of EGTI Measures
108(1)
13.5 Frozen, Liquid, and Gas phases
109(4)
14 Lecture 14: Inverse Kasteleyn Matrix for Trapezoids
113(7)
15 Lecture 15: Steepest Descent Method for Asymptotic Analysis
120(6)
15.1 Setting for Steepest Descent
120(1)
15.2 Warm-Up Example: Real Integral
120(1)
15.3 One-Dimensional Contour Integrals
121(2)
15.4 Steepest Descent for a Double Contour Integral
123(3)
16 Lecture 16: Bulk Local Limits for Tilings of Hexagons
126(9)
17 Lecture 17: Bulk Local Limits Near Straight Boundaries
135(7)
18 Lecture 18: Edge Limits of Tilings of Hexagons
142(9)
18.1 Heuristic Derivation of Two Scaling Exponents
142(2)
18.2 Edge Limit of Random Tilings of Hexagons
144(5)
18.3 The Airy Line Ensemble in Tilings and Beyond
149(2)
19 Lecture 19: The Airy Line Ensemble and Other Edge Limits
151(10)
19.1 Invariant Description of the Airy Line Ensemble
151(2)
19.2 Local Limits at Special Points of the Frozen Boundary
153(1)
19.3 From Tilings to Random Matrices
154(7)
20 Lecture 20: GUE-Corners Process and Its Discrete Analogues
161(12)
20.1 Density of GUE-Corners Process
161(4)
20.2 GUE-Corners Process as a Universal Limit
165(3)
20.3 A Link to Asymptotic Representation Theory and Analysis
168(5)
21 Lecture 21: Discrete Log-Gases
173(12)
21.1 Log-Gases and Loop Equations
173(3)
21.2 Law of Large Numbers through Loop Equations
176(3)
21.3 Gaussian Fluctuations through Loop Equations
179(3)
21.4 Orthogonal Polynomial Ensembles
182(3)
22 Lecture 22: Plane Partitions and Schur Functions
185(11)
22.1 Plane Partitions
185(2)
22.2 Schur Polynomials
187(2)
22.3 Expectations of Observables
189(7)
23 Lecture 23: Limit Shape and Fluctuations for Plane Partitions
196(11)
23.1 Law of Large Numbers
196(6)
23.2 Central Limit Theorem
202(5)
24 Lecture 24: Discrete Gaussian Component in Fluctuations
207(17)
24.1 Random Heights of Holes
207(1)
24.2 Discrete Fluctuations of Heights through GFF Heuristics
208(4)
24.3 Approach through Log-Gases
212(3)
24.4 Two-Dimensional Dirichlet Energy and One-Dimensional Logarithmic Energy
215(7)
24.5 Discrete Component in Tilings on Riemann Surfaces
222(2)
25 Lecture 25: Sampling Random Tilings
224(13)
25.1 Markov Chain Monte Carlo
224(4)
25.2 Coupling from the Past
228(4)
25.3 Sampling through Counting
232(1)
25.4 Sampling through Bijections
232(1)
25.5 Sampling through Transformations of Domains
233(4)
References 237(12)
Index 249
Vadim Gorin is a faculty member at the University of WisconsinMadison and a member of the Institute for Information Transmission Problems at the Russian Academy of Sciences. He is a leading researcher in the area of integrable probability, and has been awarded several prizes, including the Sloan Research Fellowship and the Prize of the Moscow Mathematical Society.
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