The emergence of Logarithmic-periodic oscillations in Contact process with the topological disorder

We present a model of contact process on Domany-Kinzel cellular automata with a geometrical disorder. In the 1-D model, each site is connected to two nearest neighbors which are either on the left or the right. The system is always attracted to an absorbing state with algebraic decay of average density with a continuously varying complex exponent. The log-periodic oscillations are imposed over and above the usual power law and are clearly evident as p --> 1. This effect is purely due to an underlying topology because all sites have the same infection probability p and there is no disorder in the infection rate. An extension of this model to two and three dimensions leads to similar results. We also study a model with fixed immunization rate p0 in one and two dimensions. If p0 > ps, where ps is percolation threshold, the system always tends to an absorbing state. As infection rate p --> 1, we observe a power law decay of order parameter with complex exponent. This may be a common feature in systems where quenched disorder leads to effective fragmentation of the lattice.