Critical behavior of the contact process is studied in annealed scale-free networks by mapping it on the random walk problem. We obtain the analytic results for the critical scaling, using the event-driven dynamics approach. These results are confirmed by numerical simulations. The disorder fluctuation induced by the sampling disorder in annealed networks is also explored. Finally, we discuss over the discrepancy of the finite-size-scaling theory in annealed and quenched networks in spirit of the droplet size scale and the linking disorder fluctuation.
We present an analysis of the quasi-stationary (QS) state of the contact process (CP) on annealed scale-free networks using a mapping of the CP dynamics in a one-step processes and analyzing numerically and analytically the corresponding master equation. The relevant QS quantities determined via the master equation exhibit an excellent agreement with direct QS stochastic simulations of the CP. The high accuracy of the resulting data allows to probe the strong corrections to scaling present in both the critical and supercritical regions, corrections that mask the correct finite size scaling obt…
We present high-accuracy quasi-stationary (QS) simulations of the contact process in quenched networks, built using the configuration model with both structural and natural cutoffs. The critical behavior is analyzed in the framework of the anomalous finite size scaling which was recently shown to hold for the contact process on annealed networks. It turns out that the quenched topology does not qualitatively change the critical behavior, leading only (as expected) to a shift of the transition point. The anomalous finite size scaling holds with exactly the same exponents of the annealed case, s…
We study the critical behavior of the Ising model in annealed scale-free (SF) networks of finite system size with forced upper cutoff in degree. By mapping the model onto the weighted fully connected Ising model, we derive analytic results for the finite-size scaling (FSS) near the phase transition, characterized by the cutoff-dependent two-parameter scaling with four distinct scaling regimes, in highly heterogeneous networks. These results are essentially the same as those found for the nonequilibrium contact process in annealed SF networks, except for an additional complication due to the tr…
Experimentally there exist many materials with first-order phase transitions at finite temperature that display quantum criticality. Classically, a strain-energy density coupling is known to drive first-order transitions in compressible systems, and here we generalize this Larkin-Pikin mechanism to the quantum case. We show that if the T=0 system lies above its upper critical dimension, the line of first-order transitions ends in a "quantum annealed critical point" where zero-point fluctuations restore the underlying criticality of the order parameter. The generalized Larkin-Pikin phase diagra…
Random Threshold Networks with sparse, asymmetric connections show complex dynamical behavior similar to Random Boolean Networks, with a transition from ordered to chaotic dynamics at a critical average connectivity $K_c$. In this type of model - contrary to Boolean Networks - propagation of local perturbations (damage) depends on the in-degree of the sites. $K_c$ is determined analytically, using an annealed approximation, and the results are confirmed by numerical simulations. It is shown that the statistical distributions of damage spreading near the percolation transition obey power-laws, …
The Griffiths region that is due to rare regions and the resulting local moments in disordered itinerant quantum magnets, and its influence on the critical behavior, is considered within the framework of an effective field theory. It is shown that the local moments can be described in terms of static, annealed disorder, and the physical consequences of this description are discussed.
We study the diffusion on an annealed disordered lattice with a local dynamical reorganization of bonds. We show that the typical rearrangement time depends on the renewal rate like $t_r \sim τ^α$ with $α\neq 1$. This implies that the crossover time to normal diffusion in a slow rearrangement regime shows a critical behavior at the percolation threshold. New scaling relations for the dependence of the diffusion coefficient on the renewal rate are obtained. The derivation of scaling exponents confirms the crucial role of singly connected bonds in transport properties. These results are checked …
We investigate equilibrium statistical properties of the quantum XY spin-1/2 model in an external magnetic field when the interaction and field parts are subjected to quenched or/and annealed disorder. The randomness present in the system are termed annealed or quenched depending on the relation between two different time scales - the time scale associated with the equilibriation of the randomness and the time of observation. Within a mean-field framework, we study the effects of disorders on spontaneous magnetization, both by perturbative and numerical techniques. Our primary interest is to u…
Local magnetic moments in disordered sytems can be described in terms of annealed magnetic disorder, in addition to the underlying quenched disorder. It is shown that for noninteracting electron systems at zero temperature, the annealed disorder leads to a new mechanism, and a new universality class, for a metal-insulator transition. The transition is driven by a vanishing of the thermodynamic density susceptibility rather than by localization effects. The critical behavior near two-dimensions is determined, and the underlying physics is discussed.
Cuprates show multiphase complexity that has hindered physicists search for the mechanism of high T_c for many years. A fingerprint of electronic scale invariance has been reported recently by Fratini et al. by detecting the structural scale invariance of dopants using scanning micro x-ray diffraction. In order to shed light on critical phenomena on these materials, here we propose a stylized model capturing the essential characteristics of the superconducting-isulator transition of a highly dynamical, heterogenous granular material: the Disordered Quantum Tranverse Ising Model (DQTIM) on Anne…
We numerically investigate the Olami-Feder-Christensen model on a quenched random graph. Contrary to the case of annealed random neighbors, we find that the quenched model exhibits self-organized criticality deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling, with universal critical exponents. In addition, a power law relation between the size and the duration of an avalanche exists. We propose that this may represent the correct mean-field limit of the model rather than the annealed random neighbor version.
We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length $n$ is given by $n^{-c}ϕ(n)$ for some $1<c<2$ and slowly varying $ϕ$. Disorder is introduced by having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with hig…
We solve the ferromagnetic q-state Potts model on an inhomogeneous annealed network which mimics a random recursive graph. We find that this system has the inverted Berezinskii--Kosterlitz--Thouless (BKT) phase transition for any $q \geq 1$, including the values $q \geq 3$, where the Potts model normally shows a first order phase transition. We obtain the temperature dependences of the order parameter, specific heat, and susceptibility demonstrating features typical for the BKT transition. We show that in the entire normal phase, both the distribution of a linear response to an applied local f…
We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as $P(k)\sim k^{-λ}$. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case $λ> 5$, reproduces the zeros for the Ising model on a complete graph. For $…
We study a model for microscopic segregation in a homogeneous system of particles moving on a one-dimensional lattice. Particles tend to separate from each other, and evolution ceases when at least one empty site is found between any two particles. Motion is a mixture of diffusion to nearest-neighbour sites and long-range jumps, known as annealed small-world propagation. The long-range jump probability plays the role of the small-world disorder. We show that there is an optimal value of this probability, for which the segregation process is fastest. Moreover, above a critical probability, the …
A careful critical analysis of the complexity, at the annealed level, of the Sherrington-Kirkpatrick model has been performed. The complexity functional is proved to be always invariant under the Becchi-Rouet-Stora-Tyutin supersymmetry, disregarding the formulation used to define it. We consider two different saddle points of such functional, one satisfying the supersymmetry [A. Cavagna {\it et al.}, J. Phys. A {\bf 36} (2003) 1175] and the other one breaking it [A.J. Bray and M.A. Moore, J. Phys. C {\bf 13} (1980) L469]. We review the previews studies on the subject, linking different perspec…
Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb-model is a simplified description of diffusion on percolation clusters, where the comb-like structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb-model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comb-like…
An anomalous mean-field solution is known to capture the non trivial phase diagram of the Ising model in annealed complex networks. Nevertheless the critical fluctuations in random complex networks remain mean-field. Here we show that a break-down of this scenario can be obtained when complex networks are embedded in geometrical spaces. Through the analysis of the Ising model on annealed spatial networks, we reveal in particular the spectral properties of networks responsible for critical fluctuations and we generalize the Ginsburg criterion to complex topologies.
We consider the equilibrium behavior of fluids imbibed in disordered mesoporous media, including their gas-liquid critical point when present. Our starting points are on the one hand a description of the fluid/solid-matrix system as a quenched-annealed mixture and on the other hand the Hierarchical Reference Theory (HRT) developed by A. Parola and L. Reatto to cope with density fluctuations on all length scales. The formalism combines liquid-state statistical mechanics and the theory of systems in the presence of quenched disorder. A straightforward implementation of the HRT to the quenched-an…
We investigate the avalanche temporal statistics of the Susceptible-Infected-Susceptible (SIS) model when the dynamics is critical and takes place on finite random networks. By considering numerical simulations on annealed topologies we show that the survival probability always exhibits three distinct dynamical regimes. Size-dependent crossover timescales separating them scale differently for homogeneous and for heterogeneous networks. The phenomenology can be qualitatively understood based on known features of the SIS dynamics on networks. A fully quantitative approach based on Langevin theor…
A road map to understand the relation between the onset of the superconducting state with the particular optimum heterogeneity in granular superconductors is to study a Random Tranverse Ising Model on complex networks with a scale-free degree distribution regularized by and exponential cutoff p(k) \propto k^{-γ}\exp[-k/ξ]. In this paper we characterize in detail the phase diagram of this model and its critical indices both on annealed and quenched networks. To uncover the phase diagram of the model we use the tools of heterogeneous mean-field calculations for the annealed networks and the most…
We study equilibrium properties of catalytically-activated $A + A \to \oslash$ reactions taking place on a lattice of adsorption sites. The particles undergo continuous exchanges with a reservoir maintained at a constant chemical potential $μ$ and react when they appear at the neighbouring sites, provided that some reactive conditions are fulfilled. We model the latter in two different ways: In the Model I some fraction $p$ of the {\em bonds} connecting neighbouring sites possesses special catalytic properties such that any two $A$s appearing on the sites connected by such a bond instantaneous…
A general approach for the description of spin systems on hierarchial lattices with coordination number $q$ as a dynamical variable is proposed. The ferromagnetic Ising model on the Bethe lattice was studied as a simple example demonstrating our method. The annealed and partly annealed versions of disorder concerned with the lattice coordination number are invented and discussed. Recurrent relations are obtained for the evaluation of magnetization. The magnetization is calculated for the particular disorder choices, $q=2,3$ and $q=3,4$. A nontrivial localization of critical point is revealed.
A salient feature of cyclically driven first-order phase transformations in crystals is their scale-free avalanche dynamics. This behavior has been linked to the presence of a classical critical point but the mechanism leading to criticality without extrinsic tuning remains unexplained. Here we show that the source of scaling in such systems is an annealed disorder associated with transformation-induced slip which co-evolves with the phase transformation, thus ensuring the crossing of a critical manifold. Our conclusions are based on a model where annealed disorder emerges in the form of a ran…
We analyse numerically thermal fluctuations of the static overlap between equilibrium configurations in a glass-forming liquid approaching the glass transition. We find that the emergence of slow dynamics near the onset temperature correlates with the development of non-Gaussian probability distributions of overlap fluctuations, measured using both annealed and quenched definitions. Below a critical temperature, a thermodynamic field conjugate to the overlap induces a first-order phase transition, whose existence we numerically demonstrate in the annealed case. These results establish that the…
We use a discrete-time formulation to study the asymmetric avalanche process [Phys. Rev. Lett. vol. 87, 084301 (2001)] on a finite ring and obtain an exact expression for the average avalanche size of particles as a function of toppling probabilities depending on parameters $μ$ and $α$. By mapping the model below and above the critical line onto driven interface problems, we show how different regimes of avalanches may lead to different types of critical interface behavior characterized by either annealed or quenched disorders and obtain exactly the related critical exponents which violate a w…
In this paper we study the annealed coupling of an Ising model with 2-dimensional causal dynamical triangulation model. After a short review of previous results, we prove the existence of the so-called critical line and derive its analytical properties and asymptotics. In addition, relations between the model and its dual are investigated.
A Non-Markovian generalization of one-dimensional Contact Process (CP) is being introduced in which every particle has an age and will be annihilated at its maximum age $τ$. There is an absorbing state phase transition which is controlled by this parameter. The model can demonstrate oscillatory behavior in its approach to the stationary state. These oscillations are also present in the mean-field approximation which is a first-order differential equation with time-delay. Studying dynamical critical exponents suggests that the model belongs to the DP universlity class.
Recently Carlon et. al. investigated the critical behavior of the pair contact process with diffusion [cond-mat/9912347]. Using density matrix renormalization group methods, they estimate the critical exponents, raising the possibility that the transition might belong to the same universality class as branching annihilating random walks with even numbers of offspring. This is surprising since the model does not have an explicit parity-conserving symmetry. In order to understand this contradiction, we estimate the critical exponents by Monte Carlo simulations. The results suggest that the trans…
We study the pair contact process with diffusion (PCPD) using Monte Carlo simulations, and concentrate on the decay of the particle density $ρ$ with time, near its critical point, which is assumed to follow $ρ(t) \approx ct^{-δ} +c_2t^{-δ_2}+...$. This model is known for its slow convergence to the asymptotic critical behavior; we therefore pay particular attention to finite-time corrections. We find that at the critical point, the ratio of $ρ$ and the pair density $ρ_p$ converges to a constant, indicating that both densities decay with the same powerlaw. We show that under the assumption $δ_2…
The fractal dimensions and the percolation exponents of the geometrical spin clusters of like sign at criticality, are obtained numerically for an Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify non-intersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for th…
We study the stationary properties of the two-dimensional pair contact process, a nonequilibrium lattice model exhibiting a phase transition to an absorbing state with an infinite number of configurations. The critical probability and static critical exponents are determined via Monte Carlo simulations, as well as order-parameter moment ratios and the scaling of the initial density decay. The static critical properties are consistent with the directed percolation universality class.
We calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalisations of the master operator of the one-dimensional basic contact process. Perusal of the power method together with finite-size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort. A simple reasoning leading to the appropriate choice of the microscopic time scale for time-dependent simulations of Markov chains within the so called quantum chain formulation is discussed. Our approach is applicable to any …
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.
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 u…
The supercritical series expansion of the survival probability for the one-dimensional contact process in heterogeneous and disordered lattices is used for the evaluation of the loci of critical points and critical exponents $β$. The heterogeneity and disorder are modeled by considering binary regular and irregular lattices of nodes characterized by different recovery rates and identical transmission rates. Two analytical approaches based on Nested Padé approximants and Partial Differential approximants were used in the case of expansions with respect to two variables (two recovery rates) for …
We analyze the critical behavior of a class of discrete opinion models in the presence of disorder. Within this class, each agent opinion takes a discrete value ($\pm 1$ or 0) and its time evolution is ruled by two terms, one representing agent-agent interactions and the other the degree of conviction or persuasion (a self-interaction). The mean-field limit, where each agent can interact evenly with any other, is considered. Disorder is introduced in the strength of both interactions, with either quenched or annealed random variables. With probability $p$ (1-$p$), a pairwise interaction reflec…
We investigate the role of clustering on the critical behavior of the contact process (CP) on small-world networks using the Watts-Strogatz (WS) network model with an edge rewiring probability p. The critical point is well predicted by a homogeneous cluster-approximation for the limit of vanishing clustering (p close to 1). The critical exponents and dimensionless moment ratios of the CP are in agreement with those predicted by the mean-field theory for any p > 0. This independence on the network clustering shows that the small-world property is a sufficient condition for the mean-field theory…
The critical behavior of the contact process (CP) in heterogeneous periodic and weakly-disordered environments is investigated using the supercritical series expansion and Monte Carlo (MC) simulations. Phase-separation lines and critical exponents $β$ (from series expansion) and $η$ (from MC simulations) are calculated. A general analytical expression for the locus of critical points is suggested for the weak-disorder limit and confirmed by the series expansion analysis and the MC simulations. Our results for the critical exponents show that the CP in heterogeneous environments remains in the …
The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold $x_c$ to be 1/(<k>_f+1), where <k>_f=<k^2>/<k> (=<k>) in the quenched (annealed) updating case, where <k^n> is the n-th moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent γ<3 (γ> 3) for the quenched case in the thermodynamic limit. The theoretical value x_c fits well to the numerical simulation data in the annealed case…
We propose a degree-based coarse graining approach that not just accelerates the evaluation of dynamics on complex networks, but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows. For the Ising model and susceptible-infected-susceptible epidemic model, we introduce these required conditions explicitly and further prove that they are satisfied by our coarse-grained network construction within the annealed network approximation. Finally, we numerically show that the phase transitions and fluctuations on the coarse-grained …
The challenges presented by the COVID-19 epidemic have created a renewed interest in the development of new methods to combat infectious diseases. A prominent property of the SARS-CoV-2 transmission is the significant fraction of asymptomatic transmission. This may influence the effectiveness of the standard contact tracing procedure for quarantining potentially infected individuals. However, the effects of asymptomatic transmission on the epidemic threshold of epidemic spreading on networks are largely unknown. Here we study the critical percolation transition in a simple epidemic network mod…
