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

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 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 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 et al. [Phys. Rev. E63, 046116 (2001)].

Crossover scaling in the Domany–Kinzel cellular automaton

“…[29,30,31,32]). Within a one-site approximation the above presented reaction scheme leads to the differential equation for the density of active (occupied) sites (see [4] for a detailed consideration)…”

Evolutionary games on graphs

“…Variants of this method can be used to study stationary states (evolved from a random initial state) for deterministic or stochastic cellular automata (Gutowitz et al, 1987;Atman et al, 2003). In this case one should derive a set of recursion equations for the configuration probabilities at a discrete time t +1 as a function of the configuration probabilities at time t. Using the above described construction of configuration probabilities we can obtain a finite set of equations whose stationary solution can be found analytically or numerically.…”

An Investigation of the Phase Transitions of a Family of Probabilistic Automata

“…For α < 1/3 − , the stationary distribution will have two peaks at the attracting fix-points of u(p) when n → ∞, and hence lim n→∞ H(p(n)) remains small. But for α > 1/3 − , the fix-point 0.5 is an attractor, and hence lim n→∞ H(p(n)) = log (2). Taking the results together, we get lim →0 lim n→∞ H (p(n, , α)…”