Random growth models: shape and convergence rate

TL;DR

This paper explores random growth models, focusing on their shape, convergence rates, and key properties like asymptotic shapes and fluctuations, with special attention to percolation-based models and the KPZ scaling relation.

Contribution

It provides a comprehensive treatment of shape theorems, ergodic properties, and convergence rates in random growth models, including new insights into KPZ scaling.

Findings

Shape theorem and asymptotic shape properties established

Convergence rates and scaling exponents discussed

Connections to KPZ universality class outlined

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.