Lectures on Random Lozenge Tilings

Vadim Gorin is a faculty member at the University of Wisconsin–Madison 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.

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.