Volume: 68; 2017; 161 pp; Softcover
MSC: Primary 60; 82;
Print ISBN: 978-1-4704-4183-8
Product Code: ULECT/68
List Price: $44.00
AMS Member Price: $35.20
MAA Member Price: $39.60
Electronic ISBN: 978-1-4704-4356-6
Product Code: ULECT/68.E
List Price: $44.00
AMS Member Price: $35.20
MAA Member Price: $39.60
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Supplemental Materials
50 Years of First-Passage Percolation
Share this pageAntonio Auffinger; Michael Damron; Jack Hanson
First-passage percolation (FPP) is a fundamental
model in probability theory that has a wide range of applications to
other scientific areas (growth and infection in biology, optimization
in computer science, disordered media in physics), as well as other
areas of mathematics, including analysis and geometry. FPP was
introduced in the 1960s as a random metric space. Although it is
simple to define, and despite years of work by leading researchers,
many of its central problems remain unsolved.
In this book, the authors describe the main results of FPP, with
two purposes in mind. First, they give self-contained proofs of
seminal results obtained until the 1990s on limit shapes and
geodesics. Second, they discuss recent perspectives and directions
including (1) tools from metric geometry, (2) applications of
concentration of measure, and (3) related growth and competition
models. The authors also provide a collection of old and new open
questions. This book is intended as a textbook for a graduate course
or as a learning tool for researchers.
Readership
Graduate students and researchers interested in probability theory and applications to statistical physics.
Table of Contents
Table of Contents
50 Years of First-Passage Percolation
- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Chapter 1. Introduction 18
- Chapter 2. The time constant and the limit shape 512
- 2.1. Subadditivity and the time constant 512
- 2.2. The time constant through a homogenization problem 1017
- 2.3. The limit shape: Cox-Durrett shape theorem 1118
- 2.4. Other shape theorems 1623
- 2.5. The class of possible limit shapes 1724
- 2.6. The subadditive ergodic theorem revisited 2027
- 2.7. Gromov-Hausdorff convergence 2229
- 2.8. Strict convexity of the limit shape 2532
- 2.9. Simulations 2734
- Chapter 3. Fluctuations and concentration bounds 2936
- 3.1. Variance bounds 2936
- 3.2. Logarithmic improvement to variance upper bound 3239
- 3.3. Logarithmic improvement to variance lower bound 3946
- 3.4. Concentration bounds 4552
- 3.5. Convergence of the mean for subadditive ergodic processes 5057
- 3.6. Large deviations 6168
- 3.7. Cases where Gaussian fluctuations appear 6875
- Chapter 4. Geodesics 7380
- Chapter 5. Busemann functions 109116
- 5.1. Basics of Busemann functions 109116
- 5.2. Hoffman’s argument for multiple geodesics 110117
- 5.3. Directions of geodesics via Busemann functions 112119
- 5.4. Busemann increment distributions and geodesic graphs 114121
- 5.5. Busemann functions along boundaries in ℤ² 121128
- 5.6. Nonexistence of bigeodesics in fixed directions 124131
- Chapter 6. Growth and competition models 131138
- Chapter 7. Variants of FPP and related models 141148
- Chapter 8. Summary of open questions 151158
- Bibliography 155162
- Back Cover Back Cover1169