Contact process

Abstract: We present an analysis of the classical contact process on scale-free networks. A mean-field study, both for finite and infinite network sizes, yields an absorbing-state phase transition at a finite critical value of the control parameter, characterized by a set of exponents depending on the network structure. Since finite size effects are large and the infinite network limit cannot be reached in practice, a numerical study of the transition requires the application of finite size scaling theory. Contrary to other critical phenomena studied previously, the contact process in scale-free networks exhibits a nontrivial critical behavior that cannot be quantitatively accounted for by mean-field theory.

Abstract: A long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases. The approximate densities of occupied sites, under suitable tine scaling, converge to continuous space time densities which solve stochastic p.d.e.'s. For the contact process the limiting equation is the Kolmogorov-Petrovskii-Piscuinov equation driven by branching white noise. For the voter process the limiting equation is the heat equation driven by Fisher-Wright white noise.

Abstract: We determine the first through fourth moments of the order parameter, and various ratios, for several one- and two-dimensional models with absorbing-state phase transitions. We perform a detailed analysis of the system-size dependence of these ratios and confirm that they are indeed universal for three models, the contact process, the $A$ model, and the pair contact process, belonging to the directed percolation universality class. Our studies also yield a refined estimate for the critical point of the pair contact process.