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arXiv:cond-mat/0610505 · Journal of Statistical Mechanics Theory and Experiment

Multicritical behaviour of the diluted contact process

Sílvio R. Dahmen, Lionel Sittler, Haye Hinrichsen B

Published Jan 9, 2007• 14 citations
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Abstract

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.

DOI
10.1088/1742-5468/2007/01/p01011

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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.
The contact process is a simple infection spreading model showcasing an out-of-equilibrium phase transition between a macroscopically active and an inactive phase. Such absorbing state phase transitions are often sensitive to the presence of quenched disorder. Traditionally, a phase transition in the disordered contact process is either triggered by dilution or by locally varying the infection rate. However, when both factors play an important role, a multicritical point emerges that remains poorly understood. Here, we study the multicritical contact process by large-scale Monte Carlo simulations in two and three dimensions. The multicritical behavior is found to be universal and exhibits ultra-slow, activated dynamical scaling, with exponents consistent with those predicted by the strong disorder renormalization group method. This finding indicates that the multicritical contact process belongs to the same universality class as the multicritical quantum Ising model, in both two and three dimensions.
We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to 10;{10} and system sizes up to 8000x8000 sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.
We show that the interplay between geometric criticality and dynamical fluctuations leads to a novel universality class of the contact process on a randomly diluted lattice. The nonequilibrium phase transition across the percolation threshold of the lattice is characterized by unconventional activated (exponential) dynamical scaling and strong Griffiths effects. We calculate the critical behavior in two and three space dimensions, and we also relate our results to the recently found infinite-randomness fixed point in the disordered one-dimensional contact process.
Abstract We have studied the critical properties of the contact process on a square lattice with quenched site dilution by Monte Carlo simulations. This was achieved by generating in advance the percolating cluster, through the use of an appropriate epidemic model, and then by the simulation of the contact process on the top of the percolating cluster. The dynamic critical exponents were calculated by assuming an activated scaling relation and the static exponents by the usual power law behavior. Our results are in agreement with the prediction that the quenched diluted contact process belongs to the universality class of the random transverse-field Ising model. We have also analyzed the model and determined the phase diagram by the use of a mean-field theory that takes into account the correlation between neighboring sites.
We study the effects of spatially inhomogeneous diffusion on the non-equilibrium phase transition in the contact process. The directed-percolation critical point in the contact process is known to be stable against the addition of a spatially uniform diffusion term. Correspondingly, we find quenched randomness in the diffusion rates to be irrelevant by power counting in the field-theory of the contact process. However, large-scale Monte Carlo simulations demonstrate that such diffusion disorder destabilizes the clean directed percolation critical point. Instead, the transition belongs to the same infinite-randomness universality class as the contact process with disorder in the infection or healing rates. To explain these results, we develop an effective model with an infinite diffusion rate; it shows that diffusion disorder generates an effective disorder in the healing rates. The same mechanism also appears in the field-theoretic description: Whereas diffusion disorder is irrelevant by power-counting, it generates standard random-mass disorder under renormalization. We discuss the validity of this mechanism for other absorbing state transitions and non-equilibrium phase transitions in general.
We employ large-scale Monte Carlo simulations to study a particle-hole symmetric site-diluted quantum rotor model in two dimensions. The ground state phase diagram of this system features two distinct quantum phase transitions between the superuid and the insulating (Mott glass) phases. They are separated by a multicritical point. The generic transition for dilutions below the lattice percolation threshold is driven by quantum uctuations while thetransition across the percolation threshold is due to the geometric uctuations of the lattice. We determine the location of the multicritical point between these two transitions and find its critical behavior. The multicritical exponents read z = 1:72(2), β / ν = 0:41(2), and γ/ν = 2:90(5). We compare our results to other quantum phase transitions in disordered systems, and we discuss experiments.