Contact process with long-range interactions: a study in the ensemble of   constant particle number
  We analyze the properties of the contact process with long-range interactions by the use of a kinetic ensemble in which the total number of particles is strictly conserved. In this ensemble, both annihilation and creation processes are replaced by an unique process in which a particle of the system chosen at random leaves its place and jumps to an active site. The present approach is particularly useful for determining the transition point and the nature of the transition, whether continuous or discontinuous, by evaluating the fractal dimension of the cluster at the emergence of the phase transition. We also present another criterion appropriate to identify the phase transition that consists of studying the system in the supercritical regime, where the presence of a "loop" characterizes the first-order transition. All results obtained by the present approach are in full agreement with those obtained by using the constant rate ensemble, supporting that, in the thermodynamic limit the results from distinct ensembles are equivalent. 
Carlos E. Fiore and M\'ario J. de Oliveira