Random Growth Models

Irregular and stochastic growth is all around us: tumors, bacterial colonies, infections, fluid spreading in a porous medium, propagating flame fronts. The study of simplified mathematical models of stochastic growth began in probability theory half a century ago. Quite serendipitously these models have turned out to be extremely hard to analyze. They have inspired innovative probability theory and have led to new connections between probability and other parts of mathematics. This brief overview discusses two classes of such mathematical models, namely undirected first-passage percolation (FPP) and directed last-passage percolation (LPP) on the d-dimensional integer lattice Zd. The basic idea is the following. An infection starts at the origin and progresses along nearest-neighbor lattice paths. Depending on the model, admissible paths are either directed, so that each step is forced to be one of the standard basis vectors ei, or undirected. The time it takes for the infection to reach a given lattice point is determined by random passage times assigned either to edges or to vertices of the lattice. FPP seeks the path of minimal passage time, while LPP maximizes passage time. In (undirected) first-passage percolation, we give each nearest-neighbor edge e of the lattice Zd a nonnegative randompassage time te. Collectively the random variables {te} are typically independent and identically distributed. The model is parametrized by the common probability distribution μ of the tes. The passage time of a lattice path γ (a sequence of consecutive edges) is T(γ) = ∑e∈γ te. The passage time between points x and y is T(x,y) =