Random growth models: shape and convergence rate

Abstract

Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the typical models and the basic analytical questions and properties, like existence of asymptotic shapes, fluctuations of infection times, and relations to particle systems. We then specialize to models built on percolation (first-passage percolation and last-passage percolation) and give a self-contained treatment of the shape theorem, the subadditive ergodic theorem, and conjectured and proven properties of asymptotic shapes. We finish by discussing the rate of convergence to the limit shape, along with definitions of scaling exponents and a sketch of the proof of the KPZ scaling relation.