Lectures on Random Lozenge Tilings

This is the first book dedicated to reviewing the mathematics of random tilings of large domains on the plane.

Vadim Gorin (Author)

9781108843966, Cambridge University Press

Hardback, published 9 September 2021

200 pages
23.5 x 15.8 x 2 cm, 0.52 kg

'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

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.

Preface
1. Lecture 1: introduction and tileability
2. Lecture 2: counting tilings through determinants
3. Lecture 3: extensions of the Kasteleyn theorem
4. Lecture 4: counting tilings on a large torus
5. Lecture 5: monotonicity and concentration for tilings
6. Lecture 6: slope and free energy
7. Lecture 7: maximizers in the variational principle
8. Lecture 8: proof of the variational principle
9. Lecture 9: Euler–Lagrange and Burgers equations
10. Lecture 10: explicit formulas for limit shapes
11. Lecture 11: global Gaussian fluctuations for the heights
12. Lecture 12: heuristics for the Kenyon–Okounkov conjecture
13. Lecture 13: ergodic Gibbs translation-invariant measures
14. Lecture 14: inverse Kasteleyn matrix for trapezoids
15. Lecture 15: steepest descent method for asymptotic analysis
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
19. Lecture 19: the Airy line ensemble and other edge limits
20. Lecture 20: GUE-corners process and its discrete analogues
21. Lecture 21: discrete log-gases
22. Lecture 22: plane partitions and Schur functions
23. Lecture 23: limit shape and fluctuations for plane partitions
24. Lecture 24: discrete Gaussian component in fluctuations
25. Lecture 25: sampling random tilings
References
Index.

Subject Areas: Mathematical physics [PHU], Probability & statistics [PBT], Discrete mathematics [PBD], Information theory [GPF]