Abstract

Directed percolation (DP) is a specific model whose significance is that it shares behaviour (specifically, a set of critical phenomena, characterized by universal exponents) with a large class of other models, many of reaction-diffusion type. A famous conjecture asserts that the DP universality class contains all models with a handful of broadly defined features, such as the existence of a unique ‘absorbing state’ which the system can enter but not escape. The onset of turbulence by proliferation of puffs appears to satisfy these criteria, allowing many pre-existing results to be asserted without further reference to the underlying fluid mechanics. In these two lectures I will give a brief overview of the DP class, and some of its properties. I will start from discrete models but soon move onto coarse-grained ones described by stochastic PDEs. These can be used to make specific predictions concerning the critical exponents, including their exact values in spatial dimensions d > 4 and, via the renormalization group (RG), estimates for d < 4. I will not go into the details of how the RG works, but will outline how it explains universality: models in the same universality class differ by terms that are ‘irrelevant’ in a precisely defined sense. If time allows I will also discuss some models that superficially resemble DP but are not in its universality class. These include so-called parity-conserving DP (with two absorbing states) and conserved DP (with infinitely many).