A supercritical series analysis for the generalized contact process with diffusion

Abstract

We study a model that generalizes the CP with diffusion. An additional transition is included in the model so that at a particular point of its phase diagram a crossover from the directed percolation to the compact directed percolation class will happen. We are particularly interested in the effect of diffusion on the properties of the crossover between the universality classes. To address this point, we develop a supercritical series expansion for the ultimate survival probability and analyse this series using d-log Padé and partial differential approximants. We also obtain approximate solutions in the one-and two-site dynamical mean-field approximations. We find evidences that, at variance to what happens in mean-field approximations, the crossover exponent remains close to φ = 2 even for quite high diffusion rates, and therefore the critical line in the neighborhood of the multicritical point apparently does not reproduce the mean-field result (which leads to φ = 0) as the diffusion rate grows without bound.

Dependence of the crossover exponent with the diffusion rate in the generalized contact process model

“…Using the perturbative supercritical series expansion formalism, proposed by Dickman and Jensen [5], we obtain a very precise estimates for this exponent in a generalized contact process (CP) model, * Electronic address: wgd@gibbs.if.usp.br † Electronic address: oliveira@if.usp.br ‡ Electronic address: jstilck@if.uff.br which lead us to conjecture that φ = 2, in accordance with upper bound estimates [10] and simulational results for the Domany-Kinzel cellular automaton [11], which may indicate that this exponent is the same for similar models with paralell and sequencial updates. However, in an extended version of the generalized model (including a diffusion process) we were not be able to determine the behavior of the crossover exponent φ as a function of the diffusion rate in the limit of high diffusion rate [12]. Our results show only that, in a region of weak diffusion, the crossover exponent still is φ = 2.…”

“…In a preliminar work [12], using perturbative series approach, we determine the value φ ≈ 2 up to D ≈ 0.3 and due to limitations in the series derivation, we were not able to reach larger values for the diffusion rate.…”

“…In a recent study [12], we have derived solutions for the cluster dynamics approximation in the one-, two-and threesite level. In the first case (simple mean-field approximation) the critical line is p a = 1/2, independently of the diffusion rate.…”

“…This work is organized as follows. In section II, we present the model and discuss mean-field and perturbative series results of the reference [12]. In section III we describe the numerical simulational scheme and show the new results that we obtained through it.…”

Simulational study for the crossover in the generalized contact process with diffusion

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In a recent work, Dantas and Stilck studied a model that generalizes the contact process model with diffusion [1]. Our approach, based on the supercritical expansion, showed that for a weak diffusion regime the crossover exponent between the directed percolation and compact directed percolation universality classes was φ ≈ 2.

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