**Proceedings of Symposia in Pure Mathematics**

Volume: 91;
2016;
471 pp;
Hardcover

MSC: Primary 60; 82; 05;

Print ISBN: 978-1-4704-2248-6

Product Code: PSPUM/91

List Price: $120.00

Individual Member Price: $96.00

**Electronic ISBN: 978-1-4704-2883-9
Product Code: PSPUM/91.E**

List Price: $120.00

Individual Member Price: $96.00

# Probability and Statistical Physics in St. Petersburg

Share this page *Edited by *
*V. Sidoravicius; S. Smirnov*

This book brings a reader to the cutting edge of several important directions of the contemporary probability theory, which in many cases are strongly motivated by problems in statistical physics. The authors of these articles are leading experts in the field and the reader will get an exceptional panorama of the field from the point of view of scientists who played, and continue to play, a pivotal role in the development of the new methods and ideas, interlinking it with geometry, complex analysis, conformal field theory, etc., making modern probability one of the most vibrant areas in mathematics.

#### Readership

Graduate students and research mathematicians interested in recent developments in probability and statistical physics and random structures in mathematics.

# Table of Contents

## Probability and Statistical Physics in St. Petersburg

- Cover Cover11
- Title page iii2
- Contents iii4
- Preface v6
- Ergodic theory of the Burgers equation 18
- 1. Introduction 18
- 2. Stability in stochastic dynamics 18
- 3. Ergodic theory for the Navier–Stokes system with random forcing 512
- 4. Basics on the Burgers equation 714
- 5. The Burgers equation with random forcing in compact setting 1118
- 6. The Burgers equation with Poissonian forcing 1320
- 7. Quasi-compact case 1926
- 8. Main results for space-time stationary Poissonian forcing 2229
- 9. Optimal action asymptotics and the shape function 2532
- 10. Concentration inequality for optimal action 2936
- 11. One-sided minimizers: existence, uniqueness, and coalescence 3744
- 12. Busemann functions and stationary solutions of the Burgers equation 4350
- 13. Stationary solutions: uniqueness and basins of attraction 4653
- 14. Conclusion 4754
- References 4754

- Critical point and duality in planar lattice models 5158
- Motivation 5158
- 1. “One model to rule them all” 5259
- 2. Existence of a critical point for FK percolation with 𝑞≥1 6572
- 3. A first computation of 𝑝_{𝑐} based on duality 7077
- 4. A second computation of 𝑝_{𝑐} based on integrability 8592
- 5. The phase diagram of FK percolation on \bbZ² 9299
- References 95102

- Biased random walks on random graphs 99106
- 1. Introduction 99106
- 2. One dimensional models 104111
- 3. Biased random walk on supercritical trees 113120
- 4. Biased random walks on supercritical percolation clusters 133140
- 5. Random walks on critical trees 139146
- 6. Appendix on heavy-tailed random variables 145152
- Acknowledgements 148155
- References 149156

- Lectures on integrable probability 155162
- Preface 155162
- 1. Introduction 156163
- 2. Symmetric functions 164171
- 3. Determinantal point processes 170177
- 4. From Last Passage Percolation to Plancherel measure 177184
- 5. The Schur measures and their asymptotic behavior 180187
- 6. The Schur processes and Markov chains 188195
- 7. Macdonald polynomials and directed polymers 200207
- References 210217

- Markov loops in discrete spaces 215222
- Random matrices and the Potts model on random graphs 273280
- Topics in Markov chains: Mixing and escape rate 303310
- Random walk problems motivated by statistical physics 331338
- Markov dynamics on the dual object to the infinite-dimensional unitary group 373380
- 1. Preface 373380
- 2. Dyson’s model 374381
- 3. The one-particle dynamics: a bilateral birth-death process 376383
- 4. The 𝑁-particle dynamics 378385
- 5. The method of intertwiners 379386
- 6. Examples 382389
- 7. Extremal characters of 𝑈(∞) and the boundary \Om_{∞} 383390
- 8. The master equation and the stationary distribution 386393
- 9. The Edrei–Voiculescu theorem 386393
- 10. The generator 389396
- 11. Summary 390397
- 12. Concluding remarks 390397
- References 391398

- Lectures on random nodal portraits 395402
- Smoothness and standardness in the theory of 𝐴𝐹-algebras and in the problem on invariant measures 423430
- 1. Introduction 423430
- 2. Central measures, traces, characters, actions 425432
- 3. Definitions of smooth and non-smooth 𝐴𝐹-algebras and group actions 427434
- 4. Examples of smooth and non-smooth cases, and conjecture 429436
- 5. Standard and non-standard approximation of 𝐴𝐹-algebras: The problem 434441
- References 435442

- Branching random walks and Gaussian fields 437444
- Back Cover Back Cover1482