There has been a tremendous growth in understanding of random tilings over the past 25 years. This book, the first dedicated to the topic, caters to both beginning graduate students (or advanced undergraduates) wanting to learn the basics and to mature researchers looking to widen their background knowledge.
There has been a tremendous growth in understanding of random tilings over the past 25 years. This book, the first dedicated to the topic, caters to both beginning graduate students (or advanced undergraduates) wanting to learn the basics and to mature researchers looking to widen their background knowledge.
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.
Inhaltsangabe
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.
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.
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