Scaling of spreading in models unidirectionally coupled to source particles.
We investigate the spreading behavior of evolving clusters using unidirectionally coupled two-level hierarchies in one spatial dimension. In the hierarchy, while only two source particles (A) hop away from each other without branching its offspring on the bottom level, different species of particles (B) evolve according to given dynamics belonging to one of known universality classes on the top level. Two levels are unidirectionally coupled from the bottom to the top level by the branching A-->A+2B. We derive the spreading exponent zU of the uncoupled region of size RU(t) approximately tzU up to the first order correction in terms of the spreading exponent of source particles (zA) and that of given dynamics of the top level (zo) as zU=(1--zA)zo/(1-zo). From the relation, zA and zU always satisfy the inequality zU< or =zA for zA> or =zo. The inequality confirms that the scaling of the spreading in the slave level should follow the scaling of the source in unidirectionally coupled systems. We numerically confirm the relation for three different B-particle dynamics; annihilating random walks, branching annihilating random walks with one and two offspring which belong to the directed percolation, and the parity conserving universality class respectively.
Department of Physics and Research Institute of Basic Sciences, Kyung Hee University, Seoul 130-701, Korea.
Sungchul
S
Kwon
Yup
Y
Kim