Quasi-stationary distributions for the Domany-Kinzel stochastic cellular automaton

TL;DR

This paper constructs and analyzes the quasi-stationary distributions of the Domany-Kinzel stochastic cellular automaton, revealing scaling properties similar to related processes and providing insights into its critical behavior.

Contribution

It introduces the construction of QS distributions for DKCA at one- and two-site levels and examines their scaling properties near the critical line.

Findings

QS distributions characterized by mean and moments

Scaling properties of QS state similar to contact process

Relaxation time and lifetime of QS state analyzed

Abstract

We construct the {\it quasi-stationary} (QS) probability distribution for the Domany-Kinzel stochastic cellular automaton (DKCA), a discrete-time Markov process with an absorbing state. QS distributions are derived at both the one- and two-site levels. We characterize the distribuitions by their mean, and various moment ratios, and analyze the lifetime of the QS state, and the relaxation time to attain this state. Of particular interest are the scaling properties of the QS state along the critical line separating the active and absorbing phases. These exhibit a high degree of similarity to the contact process and the Malthus-Verhulst process (the closest continuous-time analogs of the DKCA), which extends to the scaling form of the QS distribution.