Lectures on Random Lozenge Tilings

ISBN-13: 9781108843966 / Angielski / Twarda / 2021 / 200 str.

Vadim Gorin

Opis

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

Recenzje

'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

Spis treści

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.