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arXiv:0705.0317 · Journal of Statistical Mechanics Theory and Experiment

Revisiting the one-dimensional diffusive contact process

W. G. Dantas, Mário J. de Oliveira, Jürgen F. Stilck

Published Aug 1, 2007• 3 citations
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Abstract

In this work we study the one-dimensional contact process with diffusion using two different approaches to estimate the critical properties of this model: the supercritical series expansion and finite size exact solution approaches. We place special emphasis on looking at the multicritical point and its crossover exponent that characterizes the passage between directed percolation and mean-field critical properties. This crossover occurs in the limit of infinite diffusion rate and our results indicate that ϕ = 3 is a consistent result if one is sufficiently close to the multicritical point, although ϕ = 4 is the most frequent result for the curves in the region close, but not very close, to the multicritical point. This last result is consistent with the simulation data.

DOI
10.1088/1742-5468/2007/08/p08009

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In this work we study the one-dimensional contact process with diffusion using two different approaches to estimate the critical properties of this model: the supercritical series expansion and finite size exact solution approaches. We place special emphasis on looking at the multicritical point and its crossover exponent that characterizes the passage between directed percolation and mean-field critical properties. This crossover occurs in the limit of infinite diffusion rate and our results indicate that ϕ = 3 is a consistent result if one is sufficiently close to the multicritical point, although ϕ = 4 is the most frequent result for the curves in the region close, but not very close, to the multicritical point. This last result is consistent with the simulation data.
We investigate the one-dimensional pair contact process with diffusion (PCPD) by extensive Monte Carlo simulations, mainly focusing on the critical density decay exponent δ. To obtain an accurate estimate of δ, we first find the strength of corrections to scaling using the recently introduced method [S.-C. Park. J. Korean Phys. Soc. 62, 469 (2013)KPSJAS0374-488410.3938/jkps.62.469]. For small diffusion rate (d≤0.5), the leading corrections-to-scaling term is found to be ∼t^{-0.15}, whereas for large diffusion rate (d=0.95) it is found to be ∼t^{-0.5}. After finding the strength of corrections to scaling, effective exponents are systematically analyzed to conclude that the value of critical decay exponent δ is 0.173(3) irrespective of d. This value should be compared with the critical decay exponent of the directed percolation, 0.1595. In addition, we study two types of crossover. At d=0, the phase boundary is discontinuous and the crossover from the pair contact process to the PCPD is found to be described by the crossover exponent ϕ=2.6(1). We claim that the discontinuity of the phase boundary cannot be consistent with the theoretical argument supporting the hypothesis that the PCPD should belong to the DP. At d=1, the crossover from the mean field PCPD to the PCPD is described by ϕ=2 which is argued to be exact.
Crossover behaviors from the pair contact process with diffusion (PCPD) and the driven PCPD (DPCPD) to the directed percolation (DP) are studied in one dimension by introducing a single particle annihilation and/or branching dynamics. The crossover exponents phi are estimated numerically as 1/phi approximately 0.58 +/- 0.03 for the PCPD and 1/phi approximately 0.49+/-0.02 for the DPCPD. Nontriviality of the PCPD crossover exponent strongly supports the non-DP nature of the PCPD critical scaling, which is further evidenced by the anomalous critical amplitude scaling near the PCPD point. In addition, we find that the DPCPD crossover is consistent with the mean field prediction of the tricritical DP class as expected.
The phase transition of the one-dimensional, diffusive pair contact process (PCPD) is investigated by N cluster mean-field approximations and high precision simulations. The N=3,4 cluster approximations exhibit smooth transition line to absorbing state by varying the diffusion rate D with beta_2=2 mean-field order parameter exponent of the pair density. This contradicts with former N=2 results, where two different mean-field behavior was found along the transition line. Extensive dynamical simulations on L=10^5 lattices give estimates for the order parameter exponents of the particles for 0.05 <= D <= 0.7. These data can support former two distinct class findings. However the gap between low and high D exponents is narrower than estimated previously and the possibility for interpreting numerical data as a single class behavior with exponents alpha=0.21(1), beta=0.41(1) assuming logarithmic corrections is shown. Finite size scaling and cluster simulation results are also presented.
We study a contact process on a two-dimensional square lattice which is diluted by randomly removing bonds with probability p . For p <1/2 and varying birth rate λ the model was shown to exhibit a continuous phase transition which belongs to the universality class of strongly disordered directed percolation. The phase transition line terminates in a multicritical point at p = 1/2 and λ = λ * = 3.55(1), where the model can be interpreted as a critical directed percolation process running on a critical isotropic percolation cluster. In the present work we study the multicritical point and its neighbourhood by numerical simulations, discussing possible scaling forms which could describe the critical behaviour at the transition.
We provide finite-size scaling estimates for the dynamical critical exponent of the even parity-conserving universality class of critical behavior through exact numerical diagonalizations of the time evolution operator of an even-parity-conserving contact process. Our data seem to indicate that upon the introduction of a small diffusion rate in the process its critical behavior crosses over to that of the directed percolation universality class. A brief discussion of the many-sector decomposability of parity-conserving contact processes is presented.
It is shown that the universal critical properties of two recently introduced coupled directed percolation processes can be described by a single rapidity reversal invariant stochastic reaction-diffusion model. It is demonstrated that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the Gribov process (Reggeon field theory). Consequently the crossover exponent describing the scaling of the linear coupling parameter is given by Phi = 1 to all orders of the perturbation expansion.