Phase diagram of a probabilistic cellular automaton with three-site interactions

TL;DR

This paper investigates a probabilistic cellular automaton with three-site interactions, analyzing its phase transitions, critical behavior, and damage spreading, revealing a rich phase diagram with tricritical points and reentrant phases.

Contribution

It introduces a three-site interaction model extending the Domany-Kinzel automaton and maps its phase diagram, including tricritical points and damage spreading transitions, using mean-field and simulation methods.

Findings

Identified a line of tricritical points in the parameter space.

Revealed reentrant damage spreading phase for p3=0.

Reproduced known phase transitions and bicritical points for p3=1.

Abstract

We study a (1+1) dimensional probabilistic cellular automaton that is closely related to the Domany-Kinzel (DKCA), but in which the update of a given site depends on the state of {\it three} sites at the previous time step. Thus, compared with the DKCA, there is an additional parameter, $p_3$, representing the probability for a site to be active at time $t$, given that its nearest neighbors and itself were active at time $t-1$. We study phase transitions and critical behavior for the activity {\it and} for damage spreading, using one- and two-site mean-field approximations, and simulations, for $p_3=0$ and $p_3=1$. We find evidence for a line of tricritical points in the ($p_1, p_2, p_3$) parameter space, obtained using a mean-field approximation at pair level. To construct the phase diagram in simulations we employ the growth-exponent method in an interface representation. For $p_3 =0$, the phase diagram is similar to the DKCA, but the damage spreading transition exhibits a reentrant phase. For $p_3=1$, the growth-exponent method reproduces the two absorbing states, first and second-order phase transitions, bicritical point, and damage spreading transition recently identified by Bagnoli {\it et al.} [Phys. Rev. E{\bf 63}, 046116 (2001)].