Renormalisation of 2D Cellular Automata with an Absorbing State

Abstract

We describe a real-space renormalisation scheme for non-equilibrium probabilistic cellular automata (PCA) models, and apply it to a two-dimensional binary PCA. An exact renormalisation scheme is rare, and therefore we provide a method for computing the stationary probability distribution of states for such models with which to weight the renormalisation, effectively minimising the error in the scale transformation. While a mean-field approximation is trivial, we use the principle of maximum entropy to incorporate nearest-neighbour spin-correlations in the steady-state probability distribution. In doing so we find the fixed point of the renormalisation is modified by the steady-state approximation order.