**Proceedings of Symposia in Applied Mathematics**

Volume: 75;
2018;
256 pp;
Hardcover

MSC: Primary 60; 82;

**Print ISBN: 978-1-4704-3553-0
Product Code: PSAPM/75**

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

**Electronic ISBN: 978-1-4704-4907-0
Product Code: PSAPM/75.E**

List Price: $110.00

AMS Member Price: $88.00

MAA Member Price: $99.00

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# Random Growth Models

Share this page *Edited by *
*Michael Damron; Firas Rassoul-Agha; Timo Seppäläinen*

The study of random growth models began in probability theory
about 50 years ago, and today this area occupies a central place in
the subject. The considerable challenges posed by these models have
spurred the development of innovative probability theory and opened up
connections with several other parts of mathematics, such as partial
differential equations, integrable systems, and combinatorics. These
models also have applications to fields such as computer science,
biology, and physics.

This volume is based on lectures delivered at the 2017 AMS Short
Course “Random Growth Models”, held January 2–3,
2017 in Atlanta, GA.

The articles in this book give an introduction to the most-studied
models; namely, first- and last-passage percolation, the Eden model of
cell growth, and particle systems, focusing on the main research
questions and leading up to the celebrated Kardar-Parisi-Zhang
equation. Topics covered include asymptotic properties of infection
times, limiting shape results, fluctuation bounds, and geometrical
properties of geodesics, which are optimal paths for growth.

#### Readership

Graduate Students and researchers interested in various models of random growth in percolation theory, cell growth, and particle systems.

# Table of Contents

## Random Growth Models

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Bibliography ix10
- Random growth models: Shape and convergence rate 112
- Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation 3950
- 1. Motivation 3950
- 2. Geodesics and infinite geodesics 4051
- 3. Geodesic trees and the curvature condition 4455
- 4. Showing 𝒩≥2 5061
- 5. Hoffman’s method and Busemann functions 5263
- 6. Directedness and Busemann functions 5566
- 7. Busemann subsequential limits and general directedness statements 5667
- References 6778

- Fluctuations in first-passage percolation 6980
- Busemann functions, geodesics, and the competition interface for directed last-passage percolation 95106
- 1. Introduction 95106
- 2. Notation 97108
- 3. Directed last-passage percolation (LPP) 97108
- 4. Connections to other models 98109
- 5. The shape function 102113
- 6. Busemann functions 108119
- 7. Queuing fixed points 111122
- 8. Geodesics 119130
- 9. The competition interface 126137
- 10. History 128139
- 11. Next: fluctuations 129140
- References 130141

- The corner growth model with exponential weights 133144
- Exactly solving the KPZ equation 203214
- 1. Introduction 203214
- 2. Mild solution to the stochastic heat equation 206217
- 3. Scaling to the KPZ equation 210221
- 4. Weak noise convergence of polymers to SHE 215226
- 5. Moments of q–TASEP via duality and Bethe ansatz 220231
- 6. Exact formulas and the replica method 225236
- 7. Unnesting the contours and moment asymptotics 231242
- 8. From moments to the 𝑞–Laplace transform Fredholm determinant 236247
- 9. 𝑞–Boson spectral theory 240251
- Appendix A. White noise and stochastic integration 246257
- Appendix B. Background on Tracy-Widom distribution 249260
- Appendix C. Asymptotic analysis 250261
- References 253264

- Index 255266
- Back Cover Back Cover1274