Géza Ódor

Griffiths Phases on Complex Networks

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Its effects on the dynamics of complex networks have hardly been studied. Aimed at filling this gap, we analyze the contact process, i.e., the simplest propagation model, with quenched disorder on complex networks. We find Griffiths phases and other rare-region effects, leading rather generically to anomalously slow (algebraic, logarithmic, …) relaxation, on Erdos-Rényi networks. Similar effects are predicted to exist for other topologies with a finite percolation threshold. More surprisingly, we find that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of topological heterogeneity in networks with finite topological dimension. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks.

Critical behavior of a lattice prey-predator model

The critical properties of a simple prey-predator model are revisited. For some values of the control parameters, the model exhibits a line of directed percolationlike transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward an infinite number of absorbing states, and the corresponding steady-state exponents are mean-field-like. The critical behavior of the special point T (bicritical point), where the two transition lines meet, belongs to a different universality class. A particular strategy for preparing the initial states used for the dynamical Monte Carlo method is devised to correctly describe the physics of the system near the second transition line. Relationships with a forest fire model with immunization are also discussed.

Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards

The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large-scale simulations via binary lattice gases and bit-coded algorithms. We confirm scaling behavior belonging to the two-dimensional Kardar-Parisi-Zhang universality class and find a surface growth exponent: β = 0.2415(15) on 2(17) × 2(17) systems, ruling out β = 1/4 suggested by field theory. The maximum speedup with respect to a single CPU is 240. The steady state has been analyzed by finite-size scaling and a growth exponent α = 0.393(4) is found. Correction-to-scaling-exponent are computed and the power-spectrum density of the steady state is determined. We calculate the universal scaling functions and cumulants and show that the limit distribution can be obtained by the sizes considered. We provide numerical fitting for the small and large tail behavior of the steady-state scaling function of the interface width.

Phase transition of a two-dimensional binary spreading model

We investigated the phase transition behavior of a binary spreading process in two dimensions for different particle diffusion strengths (D). We found that N>2 cluster mean-field approximations must be considered to get consistent singular behavior. The N=3,4 approximations result in a continuous phase transition belonging to a single universality class along the D subset (0,1) phase transition line. Large scale simulations of the particle density confirmed mean-field scaling behavior with logarithmic corrections. This is interpreted as numerical evidence supporting the bosonic field theoretical prediction that the upper critical dimension in this model is d(c)=2. The pair density scales in a similar way but with an additional logarithmic factor to the order parameter. At the D=0 end point of the transition line we found directed percolation criticality.

Critical behavior of the one-dimensional diffusive pair contact process

The phase transition of the one-dimensional diffusive pair contact process 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<or=D<or=0.7. These data may support former two distinct class findings. However, the gap between low- and high-D exponents is narrower than previously estimated and the possibility for interpreting numerical data as a single class behavior with exponents alpha=0.21(1), beta=0.40(1) assuming logarithmic corrections is shown. Finite-size scaling results are also presented.

Roughening Transition in a Model for Dimer Adsorption and Desorption.

A solid-on-solid growth model for dimer adsorption and desorption is introduced and studied numerically. The special property of the model is that dimers can desorb only at the edges of terraces. It is shown that the model exhibits a roughening transition from a smooth to a rough phase. In both phases the interface remains pinned to the bottom layer and does not propagate. Close to the transition certain critical properties are related to those of a unidirectionally coupled hierarchy of parity-conserving branching-annihilating random walks.

Phase transition classes in triplet and quadruplet reaction-diffusion models

Phase transitions of reaction-diffusion systems with site occupation restriction and with particle creation that requires n=3,4 parents, whereas explicit diffusion of single particles (A) is present are investigated in low dimensions by the mean-field approximation and simulations. The mean-field approximation of general nA-->(n+k)A, mA-->(m-l)A type of lattice models is solved and a different kind of critical behavior is pointed out. In d=2 dimensions, the 3A-->4A, 3A-->2A model exhibits a continuous mean-field type of phase transition, that implies d(c)<2 upper critical dimension. For this model in d=1 extensive simulations support a mean-field type of phase transition with logarithmic corrections unlike the recent study of Park et al. [Phys. Rev E 66, 025101 (2002)]. On the other hand, the 4A-->5A, 4A-->3A quadruplet model exhibits a mean-field type of phase transition with logarithmic corrections in d=2, while quadruplet models in one-dimensional show robust, nontrivial transitions suggesting d(c)=2. Furthermore, I show that a parity conserving model 3A-->5A, 2A--> zero in d=1 has a continuous phase transition with different kinds of exponents. These results are in contradiction with the recently suggested implications of a phenomenological, multiplicative noise Langevin equation approach and with the simulations on suppressed bosonic systems by Kockelkoren and Chaté [Phys. Rev. Lett. 90, 125701 (2003)].

Phase transition of the one-dimensional coagulation-production process

Recently an exact solution has been found by M. Henkel and H. Hinrichsen [J. Phys. A 34, 1561 (2001)] for the one-dimensional coagulation-production process: 2A-->A, AØA-->3A with equal diffusion and coagulation rates. This model evolves into the inactive phase independently of the production rate with t(-1/2) density decay law. This paper shows that cluster mean-field approximations and Monte Carlo simulations predict a continuous phase transition for higher diffusion/coagulation rates as considered by the exact solution. Numerical evidence is given that the phase transition universality agrees with that of the annihilation-fission model with low diffusions.

Aging of the (2+1)-dimensional Kardar-Parisi-Zhang model.

Extended dynamical simulations have been performed on a (2+1)-dimensional driven dimer lattice-gas model to estimate aging properties. The autocorrelation and the autoresponse functions are determined and the corresponding scaling exponents are tabulated. Since this model can be mapped onto the (2+1)-dimensional Kardar-Parisi-Zhang surface growth model, our results contribute to the understanding of the universality class of that basic system.

Pair contact process with a particle source

We study the phase diagram and critical behavior of the one-dimensional pair contact process (PCP) with a particle source using cluster approximations and extensive simulations. The source creates isolated particles only, not pairs, and so couples not to the order parameter (the pair density) but to a nonordering field, whose state influences the evolution of the order parameter. While the critical point p(c) shows a singular dependence on the source intensity, the critical exponents appear to be unaffected by the presence of the source, except possibly for a small change in beta. In the course of our paper, we obtain high-precision values for the critical exponents of the standard PCP, confirming directed-percolationlike scaling.