Power-law relaxation of spatially disordered stochastic cellular automata and directed percolation

Abstract

This paper concerns a large class of two-state stochastic cellular automata (SCA) with inhomogeneous transition probabilities, related to generalized directed percolation (DP) on lattices with random linear defects along the preferred direction. These models were previously shown to have universal critical behavior distinct from that of standard DP. Here, I analyze the relaxation to the ``vacuum,'' the only absorbing state of such SCA. Asymptotic power-law decay with a variable exponent is derived for a substantial region in the parameter space of simple D=1 models. Disordered SCA in D=2 and 3 are more complicated, but similar power-law decay is still established by means of lower and upper bounds. The long-tailed relaxation is due to rare, very slowly decaying clusters. This is similar to the mechanism that causes the ``Griffiths' phase '' in disordered spin models.