Growth exponent in the Domany-Kinzel cellular automaton

Abstract

In a roughening process, the growth exponent β describes how the roughness w grows with the time t: w ∼ t β . We determine the exponent β of a growth process generated by the spatiotemporal patterns of the one dimensional Domany-Kinzel cellular automaton. The values obtained for β shows a cusp at the frozen/active transition which permits determination of the transition line. The β value at the transition depends on the scheme used: symmetric (β ∼ 0.83) or non-symmetric (β ∼ 0.61). Using damage spreading ideas, we also determine the active/chaotic transition line; this line depends on how the replicas are updated.

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

“…We construct the BPCA phase diagram, for p 3 = 0 and p 3 = 1, using simulations of systems of up to L = 10000 sites (with periodic boundaries), applying the growth exponent method [22] to locate the transition lines. The initial condition used in the simulations is random, with half the sites occupied.…”

“…The surface-growth process is attended by kinetic roughening; the associated critical exponents can be measured [11] following the scaling concepts developed by Family and Vicsék [25]. Atman and Moreira [22] demonstrated that the growth exponent β w exhibits a cusp at criticality, and is very useful for detecting phase transitions.…”

“…It is possible to measure β w from the slope of a log − log plot of w(L, t) versus t. In the active phase, the roughness does not saturate, growing instead as w(L, t) ∼ t 1/2 , corresponding to uncorrelated growth [22], since the correlation length is finite, away from the critical point. In previous work, Atman and Moreira [22] showed that β w attains a maximum at the phase transition, and measured its value along the transition line of the DKCA. DP values for others critical exponents in the surface representation of DKCA are verified in Ref.…”

“…We extend the mean-field analysis to the pair level, and use simulations to construct the phase diagram. In simulations, we apply the growth-exponent method [22] to identify transitions. This paper is structured as follows: in Section II we define the model and its interface representation; the site and pair mean-field approximations are discussed in Section III.…”

Universality classes of chaotic cellular automata

“…More recently, this approach has been used to study other kinds of models such as cellular automata [9,10,11], sandpiles [12], and the contact process [13]. These studies are done by mapping the model into a growth process which generates an interface -the accumulation method.…”