Dynamic phase transition in the contact process with spatial disorder: Griffiths phase and complex persistence exponents

We present a model which displays the Griffiths phase, i.e., algebraic decay of density with continuously varying exponents in the absorbing phase. In the active phase, the memory of initial conditions is lost with continuously varying complex exponents in this model. This is a one-dimensional model where a fraction r of sites obey rules leading to the directed percolation class and the rest evolve according to rules leading to the compact directed percolation class. For infection probability p≤p_{c}, the fraction of active sites ρ(t)=0 asymptotically. For p>p_{c}, ρ(∞)>0. At p=p_{c}, ρ(t), the survival probability from a single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence P_{l}(∞)>0 for p≤p_{c} and P_{l}(∞)=0 for p>p_{c}. For p≥p_{s}, local persistence P_{l}(t) decays as a power law with continuously varying exponents. The persistence exponent is clearly complex as p→1. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increase with p. We note that the underlying lattice or disorder does not have a self-similar structure.