Lectures on random lozenge tilings / Vadim Gorin.

Format:

Book

Author/Creator:

Gorin, Vadim, 1986- author.

Series:

Cambridge studies in advanced mathematics ; 193.

Cambridge studies in advanced mathematics ; 193

Language:

English

Subjects (All):

Tiling (Mathematics).

Physical Description:

1 online resource (viii, 250 pages) : digital, PDF file(s).

Edition:

First edition.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

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.

Contents:

Cover

Half-title

Series information

Title page

Copyright information

Contents

Preface

1 Lecture 1: Introduction and Tileability

1.1 Preamble

1.2 Motivation

1.3 Mathematical Questions

1.4 Thurston's Theorem on Tileability

1.5 Other Classes of Tilings and Reviews

2 Lecture 2: Counting Tilings through Determinants

2.1 Approach 1: Kasteleyn Formula

2.2 Approach 2: Lindström-Gessel-Viennot Lemma

2.3 Other Exact Enumeration Results

3 Lecture 3: Extensions of the Kasteleyn Theorem

3.1 Weighted Counting

3.2 Tileable Holes and Correlation Functions

3.3 Tilings on a Torus

4 Lecture 4: Counting Tilings on a Large Torus

4.1 Free Energy

4.2 Densities of Three Types of Lozenges

4.3 Asymptotics of Correlation Functions

5 Lecture 5: Monotonicity and Concentration for Tilings

5.1 Monotonicity

5.2 Concentration

5.3 Limit Shape

6 Lecture 6: Slope and Free Energy

6.1 Slope in a Random Weighted Tiling

6.2 Number of Tilings of a Fixed Slope

6.3 Concentration of the Slope

6.4 Limit Shape of a Torus

7 Lecture 7: Maximizers in the Variational Principle

7.1 Review

7.2 The Definition of Surface Tension and Class of Functions

7.3 Upper Semicontinuity

7.4 Existence of the Maximizer

7.5 Uniqueness of the Maximizer

8 Lecture 8: Proof of the Variational Principle

9 Lecture 9: Euler-Lagrange and Burgers Equations

9.1 Euler-Lagrange Equations

9.2 Complex Burgers Equation via a Change of Coordinates

9.3 Generalization to q[sup(Volume)]-Weighted Tilings

9.4 Complex Characteristics Method

10 Lecture 10: Explicit Formulas for Limit Shapes

10.1 Analytic Solutions to the Burgers Equation

10.2 Algebraic Solutions

10.3 Limit Shapes via Quantized Free Probability

11 Lecture 11: Global Gaussian Fluctuations for the Heights.

11.1 Kenyon-Okounkov Conjecture

11.2 Gaussian Free Field

11.3 Gaussian Free Field in Complex Structures

12 Lecture 12: Heuristics for the Kenyon-Okounkov Conjecture

13 Lecture 13: Ergodic Gibbs Translation-Invariant Measures

13.1 Tilings of the Plane

13.2 Properties of the Local Limits

13.3 Slope of EGTI Measure

13.4 Correlation Functions of EGTI Measures

13.5 Frozen, Liquid, and Gas phases

14 Lecture 14: Inverse Kasteleyn Matrix for Trapezoids

15 Lecture 15: Steepest Descent Method for Asymptotic Analysis

15.1 Setting for Steepest Descent

15.2 Warm-Up Example: Real Integral

15.3 One-Dimensional Contour Integrals

15.4 Steepest Descent for a Double Contour Integral

16 Lecture 16: Bulk Local Limits for Tilings of Hexagons

17 Lecture 17: Bulk Local Limits Near Straight Boundaries

18 Lecture 18: Edge Limits of Tilings of Hexagons

18.1 Heuristic Derivation of Two Scaling Exponents

18.2 Edge Limit of Random Tilings of Hexagons

18.3 The Airy Line Ensemble in Tilings and Beyond

19 Lecture 19: The Airy Line Ensemble and Other Edge Limits

19.1 Invariant Description of the Airy Line Ensemble

19.2 Local Limits at Special Points of the Frozen Boundary

19.3 From Tilings to Random Matrices

20 Lecture 20: GUE-Corners Process and Its Discrete Analogues

20.1 Density of GUE-Corners Process

20.2 GUE-Corners Process as a Universal Limit

20.3 A Link to Asymptotic Representation Theory and Analysis

21 Lecture 21: Discrete Log-Gases

21.1 Log-Gases and Loop Equations

21.2 Law of Large Numbers through Loop Equations

21.3 Gaussian Fluctuations through Loop Equations

21.4 Orthogonal Polynomial Ensembles

22 Lecture 22: Plane Partitions and Schur Functions

22.1 Plane Partitions

22.2 Schur Polynomials

22.3 Expectations of Observables.

23 Lecture 23: Limit Shape and Fluctuations for Plane Partitions

23.1 Law of Large Numbers

23.2 Central Limit Theorem

24 Lecture 24: Discrete Gaussian Component in Fluctuations

24.1 Random Heights of Holes

24.2 Discrete Fluctuations of Heights through GFF Heuristics

24.3 Approach through Log-Gases

24.4 Two-Dimensional Dirichlet Energy and One-Dimensional Logarithmic Energy

24.5 Discrete Component in Tilings on Riemann Surfaces

25 Lecture 25: Sampling Random Tilings

25.1 Markov Chain Monte Carlo

25.2 Coupling from the Past (Propp and Wilson, 1996)

25.3 Sampling through Counting

25.4 Sampling through Bijections

25.5 Sampling through Transformations of Domains

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 01 Sep 2021).

Includes bibliographical references and index.

ISBN:

9781108922906

1108922902

9781108921183

1108921183