Analysis and Stochastics of Growth Processes and Interface Models

This book is a collection of topical survey articles by leading researchers in the fields of applied analysis and probability theory, working on the mathematical description of growth phenomena. Particular emphasis is on the interplay of the two fields, with articles by analysts being accessible for researchers in probability, and vice versa. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic multi-scale techniques and homogenisation of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose-Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe. The combination of articles from the two fields of analysis and probability is highly unusual and makes this book an important resource for researchers working in all areas close to the interface of these fields.

1.1 - Directed Random Growth Models on the Plane
1.2 - The Pleasures and Pains of Studying the Two-Type Richardson Model
1.3 - Ballistic Phase of Self-Interacting Random Walks
2.1 - Stochastic Homogenization and Energy of Infinite Sets of Points
2.2 - Validity and Non-Validity of Propagation of Chaos
3.1 - Applications of the Lace Expansion to Statistical-Mechanical Models
3.2 - Large Deviations for Empirical Cycle Counts of Integer Partitions and Their Relation to Systems of Bosons
3.3 - Interacting Brownian Motions and the Gross-Pitaevskii Formula
3.4 - A Short Introduction to Anderson Localization
4.1 - Effective Theories for Ostwald Ripening
4.2 - Switching Paths for Ising Models with Long-Range Interaction
4.3 - Nucleation and Droplet Growth as a Stochastic Process
5.1 - On the Stochastic Burgers Equation with some Applications to Turbulence and Astrophysics
5.2 - Liquid Crystals and Harmonic Maps in Polyhedral Domains

Peter Mörters is a professor of probability at the University of Bath. Receiving his PhD from the University of London in the area of geometric measure theory, his current interests focus on Bronwnian motion and random walk, stohastic processes in random environments, large deviation theory and, more recently random networks. Roger Moser is a lecturer of mathematics at the University of Bath. He received his PhD from the Eidgenössische Technische Hochschule Zurich in the area of geometric analysis. Further current research interests include the theory of partial differential equations, the calculus of variations, geometric measure theory, and applications if mathematical phsyics. Mathew Penrose is a professor of Probability at the University of Bath. His current research interests are mainly in stohastic geometry and interacting particle systems. His monograph "Random Geometric Graphs" was published by Oxford University Press in 2003. He obtained his PhD in stohastic analysis from the University of Edinburgh. Hartmut Schwetlick is a lecturer of mathematics at the University of Bath. After receiving his PhD from the University of Tubingen in the field of nonlinear transport equations, he worked on partial differential equations and their applications at ETH Zurich and the Max Planck Institute for Mathematics in teh Sciences, Leipzig. His research interests include analysis of PDE, variational methods in geometric analysis, and nonlinear elasticity. Johannes Zimmer is currently a lecturer of applied mathematics at the University of Bath.Prior to that, he was head of an Emmy Noether group at the Max Planck Institute for Mathematics in the Sciences, Leipzig. He obtained his doctorate from the Technische universitat Munchen. Research interests include the analysis of lattice dynamical systems, and PDEs, continuum mechanics, and phase transitions.