Asymmetrically coupled directed percolation systems

We introduce a dynamical model of coupled directed percolation systems with two particle species. The two species A and B are coupled asymmetrically in that A particles branch B particles, whereas B particles prey on A particles. This model may describe epidemic spreading controlled by reactive immunization agents. We study nonequilibrium phase transitions with attention focused on the multicritical point where both species undergo the absorbing phase transition simultaneously. In one dimension, we find that the inhibitory coupling from B to A is irrelevant and the model belongs to the unidirectionally coupled directed percolation class. On the contrary, a mean-field analysis predicts that the inhibitory coupling is relevant and a new universality appears with a variable dynamic exponent. Numerical simulations on small-world networks confirm our predictions.     

Nonequilibrium critical behavior in unidirectionally coupled stochastic processes

Phase transitions from an active into an absorbing, inactive state are generically described by the critical exponents of directed percolation (DP), with upper critical dimension d(c)=4. In the framework of single-species reaction-diffusion systems, this universality class is realized by the combined processes A-->A+A, A+A-->A, and A-->0. We study a hierarchy of such DP processes for particle species A,B,..., unidirectionally coupled via the reactions A-->B, ...(with rates mu(AB),...). When the DP critical points at all levels coincide, multicritical behavior emerges, with density exponents beta(i) which are markedly reduced at each hierarchy level i> or =2. This scenario can be understood on the basis of the mean-field rate equations, which yield beta(i)=1/2(i-1) at the multicritical point. Using field-theoretic renormalization-group techniques in d=4-epsilon dimensions, we identify a new crossover exponent phi, and compute phi=1+O(epsilon(2)) in the multicritical regime (for small mu(AB)) of the second hierarchy level. In the active phase, we calculate the fluctuation correction to the density exponent on the second hierarchy level, beta(2)=1/2-epsilon/8+O(epsilon(2)). Outside the multicritical region, we discuss the crossover to ordinary DP behavior, with the density exponent beta(1)=1-epsilon/6+O(epsilon(2)). Monte Carlo simulations are then employed to confirm the crossover scenario, and to determine the values for the new scaling exponents in dimensions d< or =3, including the critical initial slip exponent. Our theory is connected to specific classes of growth processes and to certain cellular automata, and the above ideas are also applied to unidirectionally coupled pair annihilation processes. We also discuss some technical as well as conceptual problems of the loop expansion, and suggest some possible interpretations of these difficulties.     

Finite size scaling for directed percolation and related stochastic evolution processes

Finite size scaling effects are investigated for certain evolution processes modelled by a one-component reaction-diffusion system with an absorbing state. The model possesses a non-equilibrium critical point, and the associated universality class includes directed bond percolation, cellular automata, Reggeon field theory and a stochastic version of Schlögl's first autocatalytic reaction scheme. Using renormalisation group techniques, we calculate the linear relaxation time in a cubic geometry of finite sizeL, with periodic boundary conditions imposed. The corresponding scaling behaviour toO() (=4–d,d being the spatial dimension) is presented in universal form.     

The field theory approach to percolation processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous non-equilibrium active to absorbing state phase transitions whose asymptotic features are governed, respectively, by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field theory representation for these Markovian stochastic processes based on fundamental phenomenological considerations, as well as from a specific microscopic reaction-diffusion model realization. Subsequently we explain how dynamic renormalization group (RG) methods can be applied to obtain the universal properties near the critical point in an expansion about the upper critical dimensions d_c = 4 (DP) and 6 (dIP). We provide a detailed overview of results for critical exponents, scaling functions, crossover phenomena, finite-size scaling, and also briefly comment on the influence of long-range spreading, the presence of a boundary, multispecies generalizations, coupling of the order parameter to other conserved modes, and quenched disorder.     

Criticality of Epidemic Spreading in Mobile Individuals Mediated by Environment

We investigate the critical behaviour of an epidemical model in a diffusive population mediated by a static vector environment on a 2D network. It is found that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. Finite-size and short-time dynamic scaling relations are used to determine the critical population density and the critical exponents characterizing the behaviour near the critical point. The results are compatible with the universality class of directed percolation coupled to a conserved diffusive field with equal diffusion constants.     

Nonlinear critical relaxation of the Ginzburg Landau field coupled to a conserved density

We show that the time-dependent Ginzburg-Landau model for a one-component order parameter coupled to a nonordering conserved density (Model C of Halperin and Hohenberg) describes the dynamic critical behavior of a certain class of ordering alloys. An equation of motion for the order parameter in scaling form is derived and used to discuss nonlinear relaxation processes that occur near the critical point after a sudden change in temperature or the symmetry-breaking field. Corresponding universal amplitude ratios are calculated to one-loop order. Unlike the dynamic critical exponent these ratios show large corrections due to fluctuations and due to the coupling to the conserved density in 4- spatial dimensions. We also verify Fisher renormalization for the model to all orders.     

Colossal magnetoresistance in manganites as a multicritical phenomenon

The colossal magnetoresistance in manganites AMnO3 is studied from the viewpoint of multicritical phenomena. To understand the complicated interplay of various phases, we study the Ginzburg-Landau theory in terms of both the mean-field approximation and the renormalization-group analysis for comparison with the observed phase diagram. Several novel features, such as the first-order ferromagnetic transition and the dip in the transition temperature near the multicritical point, can be understood as being driven by enhanced fluctuations near the multicritical point. Furthermore, we obtain a universal scaling relation for the H/M versus M2 plot (Arrott plot), which fits rather well with the experimental data, providing further evidence for the enhanced fluctuation.     

Quantum versus geometric disorder in a two-dimensional Heisenberg antiferromagnet

We present a numerical study of the spin-1/2 bilayer Heisenberg antiferromagnet with random interlayer dimer dilution. From the temperature dependence of the uniform susceptibility and a scaling analysis of the spin correlation length we deduce the ground state phase diagram as a function of nonmagnetic impurity concentration p and bilayer coupling g. At the site percolation threshold, there exists a multicritical point at small but nonzero bilayer coupling g(m)=0.15(3). The magnetic properties of the single-layer material La(2)Cu(1-p)(Zn,Mg)(p)O4 near the percolation threshold appear to be controlled by the proximity to this new quantum critical point.     

Absence of hyperscaling violations for phase transitions with positive specific heat exponent

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of generalH-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases =3, =5 and =15 where is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.     

Regularities of spectral component behavior near the phase synchronization boundary in spatially extended systems

In this work we study the dynamics of spatially coupled systems using the example of unidirectionally coupled Pierce diodes near the boundary of phase chaotic synchronization. We show that the dynamics in the investigated region of the coupling parameter obeys a universal regularity valid for different space points of the system.     

Infinite-range mean-field percolation: Transfer matrix study of longitudinal correlation length

Two infinite-range directed percolation models, equivalent also to epidemic models, are considered for a finite population (finite number of sites)N at each time (directed axis) step. The general features of the transfer matrix spectrum (evolution operator spectrum for the epidemic) are studied numerically, and compared with analytical predictions in the limitN = . One of the models is devised to allow numerical results to be obtained forN as high as nearly 800 for the largest longitudinal percolation correlation length (relaxation time for epidemic). The finite-N behavior of this correlation length is studied in detail, including scaling near the percolation transition, exponential divergence (withN) above the percolation transition, as well as other noncritical and critical-point properties.     

Multicritical point in a diluted bilayer Heisenberg quantum antiferromagnet

The S=1/2 Heisenberg bilayer antiferromagnet with randomly removed interlayer dimers is studied using quantum Monte Carlo simulations. A zero-temperature multicritical point (p(*),g(*)) at the classical percolation density p=p(*) and interlayer coupling g(*) approximately equal 0.16 is demonstrated. The quantum critical exponents of the percolating cluster are determined using finite-size scaling. It is argued that the associated finite-temperature quantum critical regime extends to zero interlayer coupling and could be relevant for antiferromagnetic cuprates doped with nonmagnetic impurities.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Universal Scaling Behavior of Directed Percolation Around the Upper Critical Dimension

In this work we consider the steady state scaling behavior of directed percolation around the upper critical dimension. In particular we determine numerically the order parameter, its fluctuations as well as the susceptibility as a function of the control parameter and the conjugated field. Additionally to the universal scaling functions, several universal amplitude combinations are considered. We compare our results with those of a renormalization group approach.     

First-Passage Percolation, Semi-Directed Bernoulli Percolation, and Failure in Brittle Materials

We present a two-dimensional, quasistatic model of fracture in disordered brittle materials that contains elements of first-passage percolation, i.e., we use a minimum-energy-consumption criterion for the fracture path. The first-passage model is employed in conjunction with a semi-directed Bernoulli percolation model, for which we calculate critical properties such as the correlation length exponent v ^sdir and the percolation threshold p _c ^sdir . Among other results, our numerics suggest that v ^sdir is exactly 3/2, which lies between the corresponding known values in the literature for usual and directed Bernoulli percolation. We also find that the well-known scaling relation between the wandering and energy fluctuation exponents breaks down in the vicinity of the threshold for semi-directed percolation. For a restricted class of materials, we study the dependence of the fracture energy (toughness) on the width of the distribution of the specific fracture energy and find that it is quadratic in the width for small widths for two different random fields, suggesting that this dependence may be universal.     

Finite-size scaling for mean-field percolation

By studying transfer matrix eigenvalues, correlation lengths for a mean field directed percolation model are obtained both near and far from the critical regime. Near criticality, finite-size scaling behavior is derived and an analytic technique is provided for obtaining the finite-size scaling function. Our methods involve the generating function, matched asymptotic expansions, and certain formulas developed for the study of eigenvalues of the transfer matrix for metastability.     

Scaling behavior of the directed percolation universality class

In this work we consider five different lattice models which exhibit continuous phase transitions into absorbing states. By measuring certain universal functions, which characterize the steady state as well as the dynamical scaling behavior, we present clear numerical evidence that all models belong to the universality class of directed percolation. Since the considered models are characterized by different interaction details the obtained universal scaling plots are an impressive manifestation of the universality of directed percolation.     

Criticality of Parasitic Disease Transmission in a Diffusive Population

Through using the methods of finite-size effect and short time dynamic scaling, we study the critical behavior of parasitic disease spreading process in a diffusive population mediated by a static vector environment. Through comprehensive analysis of parasitic disease spreading we find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. We determine the critical population density, above which the system reaches an epidemic spreading stationary state. We also perform a scaling analysis to determine the order parameter and critical relaxation exponents. The results show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.     

A higher symmetry in the sol-gel transition at the consolute point

By methods of renormalized field theory we show that generically the multicritical behavior of the sol-gel transition at the consolute point is dominated by a fixed point symmetry which is higher than the symmetry of the original Hamiltonian. In general, this leads to Ising-like behavior of all fluctuations. We show in particular that the Fisher exponent _P of the percolation field coincides with the corresponding exponent _I of Ising fields. We perform a preliminary stability analysis which indicates that the higher symmetry is not broken in the physical 3-dimensional case.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

The effect of finite response–time in coupled dynamical systems

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous (τ?=?0) case. The critical coupling strength at which synchronization sets in is found to increase with τ. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.