N. Boccara et al., CRITICAL-BEHAVIOR OF A PROBABILISTIC LOCAL AND NONLOCAL SITE-EXCHANGECELLULAR AUTOMATION, International journal of modern physics C, 5(3), 1994, pp. 537-545
We study the critical behavior of a probabilistic automata network who
se local rule consists of two subrules. The first one, applied synchro
nously, is a probabilistic one-dimensional range-one cellular automato
n rule. The second, applied sequentially, exchanges the values of a pa
ir of sites. According to whether the two sites are first-neighbors or
not, the exchange is said to be local or nonlocal. The evolution of t
he system depends upon two parameters, the probability p characterizin
g the probabilistic cellular automaton, and the degree of mixing m res
ulting from the exchange process. Depending upon the values of these p
arameters, the system exhibits a bifurcation similar to a second order
phase transition characterized by a nonnegative order parameter, whos
e role is played by the stationary density of occupied sites. When m i
s very large, the correlations created by the application of the proba
bilistic cellular automaton rule are destroyed, and, as expected, the
behavior of the system is then correctly predicted by a mean-field-typ
e approximation. According to whether the exchange of the site values
is local or nonlocal, the critical behavior is qualitatively different
as m varies.