Local and global persistence exponents associated with zero temperature quenched dynamics of two dimensional XY model and three dimensional Heisenberg model have been estimated using numerical simulations. We have used the method of block persistence to find both global and local exponents simultaneously (in a single simulation). Temperature universality of both the exponents for three dimensional Heisenberg model has been confirmed by simulating the stochastic (with noise) version of the equation of motion. The noise amplitudes added were small enough to retain the dynamics below criticality. In the second part of our work we have studied scaling associated with correlated persistence sites in the three dimensional Heisenberg model in the later stages of the dynamics. The relevant length scale associated with correlated persistent sites was found to behave in a manner similar to the dynamic length scale associated with the phase ordering dynamics.
We examine persistence in one dimensional Ising model under zero temperature Glauber dynamics for random initial states with unequal fraction of up and down spins. We find the persistence exponent varies continuously with the fraction of up spins in the initial state. Apparently this feature has been overlooked in the studies so far.
It was recently realized that the persistence exponent appearing in the dynamics of nonequilibrium systems is a special member of a continuously varying family of exponents, describing generalized persistence properties. We propose and solve a simplified model of coarsening, where time intervals between spin flips are independent, and distributed according to a Lévy law. Both the limit distribution of the mean magnetization and the generalized persistence exponents are obtained exactly.
We measure the persistence exponent in a phase separating two-dimensional spin system with non-conserved dynamics quenched in a region with four coexisting stripe phases. The system is an Ising model with nearest neighbor, next-to-the-nearest neighbor and plaquette interactions. Due the particular nature of the ground states, the order parameter is defined in terms of blocks of spins. Our estimate of the persistence exponent, $θ=0.42$, differs from those of the two-dimensional Ising and four state Potts models. Our procedure allows the study of persistence properties also at finite temperature…
Extensive simulations are performed to study the persistence behavior of a conserved lattice gas model exhibiting an absorbing phase transition from an active phase into an inactive phase. Both the global and the local persistence exponents are determined in two and higher dimensions. The local persistence exponent obeys a scaling relation involving the order parameter exponent of the absorbing phase transition. Furthermore we observe that the global persistence exponent exceeds its local counterpart in all dimensions in contrast to the known persistence behavior in reversible phase transition…
We consider the d-dimensional diffusion equation for a field phi(x,t) with random initial condition, and observe that, when appropriately scaled, phi(0,t) is Gaussian and Markovian in the limit d->0. This leads via the Majumdar-Sire perturbation theory to a small-d expansion for the persistence exponent theta(d). We find theta(d) = d/4 - 0.12065...d^{3/2} + ...
We extend the definition of a global order parameter to the case of a critical system confined between two infinite parallel plates separated by a finite distance $L$. For a quench to the critical point we study the persistence property of the global order parameter and show that there is a crossover behaviour characterized by a non universal exponent which depends on the ratio of the system size to a dynamic length scale.
We obtain θ_p(q) = 2θ_s(q) for one-dimensional q-state ferromagnetic Potts models evolving under parallel dynamics at zero temperature from an initially disordered state, where θ_p(q) is the persistence exponent for parallel dynamics and θ_s(q) = -{1/8}+ \frac{2}{π^2}[cos^{-1}{(2-q)/q\sqrt{2}}]^2 [PRL, {\bf 75}, 751, (1995)], the persistence exponent under serial dynamics. This result is a consequence of an exact, albeit non-trivial, mapping of the evolution of configurations of Potts spins under parallel dynamics to the dynamics of two decoupled reaction diffusion systems.
We consider an arbitrary Gaussian Stationary Process X(T) with known correlator C(T), sampled at discrete times T_n = n ΔT. The probability that (n+1) consecutive values of X have the same sign decays as P_n \sim \exp(-θ_D T_n). We calculate the discrete persistence exponent θ_D as a series expansion in the correlator C(ΔT) up to 14th order, and extrapolate to ΔT = 0 using constrained Padé approximants to obtain the continuum persistence exponent θ. For the diffusion equation our results are in exceptionally good agreement with recent numerical estimates.
The local persistence R(t), defined as the proportion of the system still in its initial state at time t, is measured for the Bak--Sneppen model. For 1 and 2 dimensions, it is found that the decay of R(t) depends on one of two classes of initial configuration. For a subcritical initial state, R(t)\sim t^{-θ}, where the persistence exponent θcan be expressed in terms of a known universal exponent. Hence θis universal. Conversely, starting from a supercritical state, R(t) decays by the anomalous form 1-R(t)\sim t^{τ_{\rm ALL}} until a finite time t_{0}, where τ_{\rm ALL} is also a known exponent…
The global persistence exponent $θ_g$ is calculated for the two-dimensional Blume-Capel model following a quench to the critical point from both disordered states and such with small initial magnetizations. Estimates are obtained for the nonequilibrium critical dynamics on the critical line and at the tricritical point. Ising-like universality is observed along the critical line and a different value $θ_g =1.080(4)$ is found at the tricritical point.
The persistence exponent θfor the global order parameter, M(t), of a system quenched from the disordered phase to its critical point describes the probability, p(t) \sim t^{-θ}, that M(t) does not change sign in the time interval t following the quench. We calculate θto O(ε^2) for model A of critical dynamics (and to order εfor model C) and show that at this order M(t) is a non-Markov process. Consequently, θis a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Majumdar and Sire [Phys. …
We present a model which displays Griffiths phase i.e. algebraic decay of density with continuously varying exponent in the absorbing phase. In active phase, the memory of initial conditions is lost with continuously varying complex exponent in this model. This is 1-D model where fraction r of sites obey rules leading to directed percolation (DP) class and the rest evolve according to rules leading to compact directed percolation (CDP) class. For infection probability $p < p_c$, the fraction of active sites $ρ(t) = 0$ asymptotically. For $p > p_c$, $ρ(infty) > 0$. At $p = p_c$, $ρ(t)$, the sur…
We show experimentally and theoretically that the persistence of large deviations in equilibrium step fluctuations is characterized by an infinite family of independent exponents. These exponents are obtained by carefully analyzing dynamical experimental images of Al/Si(111) and Ag(111) equilibrium steps fluctuating at high (970K) and low (320K) temperatures respectively, and by quantitatively interpreting our observations on the basis of the corresponding coarse-grained discrete and continuum theoretical models for thermal surface step fluctuations under attachment/detachment (``high-temperat…
Persistence probabilities of the interface height in (1+1)- and (2+1)-dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence expo…
We obtained the global persistence exponent $θ_g$ for a continuous spin model on the simple cubic lattice with double-exchange interaction by using two different methods. First, we estimated the exponent $θ_g$ by following the time evolution of probability $P(t)$ that the order parameter of the model does not change its sign up to time $t$ $[P(t)\thicksim t^{-θ_g}]$. Afterwards, that exponent was estimated through the scaling collapse of the universal function $L^{θ_g z} P(t)$ for different lattice sizes. Our results for both approaches are in very good agreement each other.
We discuss the interplay between the degree of dynamical stochasticity, memory persistence and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasi-deterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including mean-field, and the sho…
Persistence in coarsening 1D spin systems with a power law interaction $r^{-1-σ}$ is considered. Numerical studies indicate that for sufficiently large values of the interaction exponent $σ$ ($σ\geq 1/2$ in our simulations), persistence decays as an algebraic function of the length scale $L$, $P(L)\sim L^{-θ}$. The Persistence exponent $θ$ is found to be independent on the force exponent $σ$ and close to its value for the extremal ($σ\to \infty$) model, $\barθ=0.17507588...$. For smaller values of the force exponent ($σ< 1/2$), finite size effects prevent the system from reaching the asymptoti…
We obtain the persistence exponents for an antiferromagnetic Ising system in which the magnetisation is kept constant. This system belongs to Model C (system with non-conserved order parameter with a conserved density) and is expected to have persistence exponents different from that of Model A (system with no conservation) but independent of the conserved density. Our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimension…
We study persistence in coupled circle map lattices at the onset of spatiotemporal intermittency, an onset which marks a continuous transition, in the universality class of directed percolation, to a unique absorbing state. We obtain a local persistence exponent of theta_l = 1.49 +- 0.02 at this transition, a value which closely matches values for theta_l obtained in stochastic models of directed percolation. This result constitutes suggestive evidence for the universality of persistence exponents at the directed percolation transition. Given that many experimental systems are modelled accurat…
The question of universality class of pair contact process with diffusion (PCPD) is revisited with an alternative approach. We study persistence in Generalized Pair-Contact Process with diffusion (GPCPD) introduced by Noh and Park, (Phys. Rev. E 69,016122(2004)). This model allows us to interpolate between directed percolation (DP) and PCPD universality classes. We find that transition to nonzero persistence is at same parameter value as transition to zero number density. We obtain finite size scaling and off-critical scaling collapse for persistence and find critical exponents by fitting phen…
Motivated by certain problems of statistical physics we consider a stationary stochastic process in which deterministic evolution is interrupted at random times by upward jumps of a fixed size. If the evolution consists of linear decay, the sample functions are of the "random sawtooth" type and the level dependent persistence exponent θcan be calculated exactly. We then develop an expansion method valid for small curvature of the deterministic curve. The curvature parameter g plays the role of the coupling constant of an interacting particle system. The leading order curvature correction to θi…
Numerically we solve the microscopic deterministic equations of motion with random initial states for the two-dimensional $ϕ^4$ theory. Scaling behavior of the persistence probability at criticality is systematically investigated and the persistence exponent is estimated.
For the two dimensional kinetic Ising model at finite temperature, the local mean magnetisation $M_t=t^{-1}\int_{0}^tσ(t')\d t'$, simply related to the fraction of time spent by a given spin in the positive direction, has a limiting distribution, singular at $\pm m_0(T)$, the Onsager spontaneous magnetization. The exponent of this singularity defines the persistence exponent $θ$. We also study first passage exponents associated to persistent large deviations of $M_t$, and their temperature dependence.
We study the persistence P(t) of the magnetization of a d' dimensional manifold (i.e., the probability that the manifold magnetization does not flip up to time t, starting from a random initial condition) in a d-dimensional spin system at its critical point. We show analytically that there are three distinct late time decay forms for P(t) : exponential, stretched exponential and power law, depending on a single parameter ζ=(D-2+η)/z where D=d-d' and η, z are standard critical exponents. In particular, our theory predicts that the persistence of a line magnetization decays as a power law in the…
We introduce a parameter $p$, called partial survival, in the persistence of stochastic processes and show that for smooth processes the persistence exponent $θ(p)$ changes continuously with $p$, $θ(0)$ being the usual persistence exponent. We compute $θ(p)$ exactly for a one-dimensional deterministic coarsening model, and approximately for the diffusion equation. Finally we develop an exact, systematic series expansion for $θ(p)$, in powers of $ε=1-p$, for a general Gaussian process with finite density of zero crossings.
We consider a directed percolation process at its critical point. The probability that the deviation of the global order parameter with respect to its average has not changed its sign between 0 and t decays with t as a power law. In space dimensions d<4 the global persistence exponent theta_p that characterizes this decay is theta_p=2 while for d<4 its value is increased to first order in epsilon = 4-d. Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to theta_p to first order in epsilon. The global persistence exponent is found …
The persistence properties of a set of random walkers obeying the A+B -> 0 reaction, with equal initial density of particles and homogeneous initial conditions, is studied using two definitions of persistence. The probability, P(t), that an annihilation process has not occurred at a given site has the asymptotic form $P(t) -> const + t^{-θ}$, where $θ$ is the persistence exponent (``type I persistence''). We argue that, for a density of particles $ρ>> 1$, this non-trivial exponent is identical to that governing the persistence properties of the one-dimensional diffusion equation, where $θ\appr…
For many stochastic processes, the probability $S(t)$ of not-having reached a target in unbounded space up to time $t$ follows a slow algebraic decay at long times, $S(t)\sim S_0/t^θ$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $θ$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ for a compact random walk in unbounded space by establishing an analytic relation with the mean first…
We show that the persistence probability $P(t,L)$, in a coarsening system of linear size $L$ at a time $t$, has the finite size scaling form $P(t,L)\sim L^{-zθ}f(\frac{t}{L^{z}})$ where $θ$ is the persistence exponent and $z$ is the coarsening exponent. The scaling function $f(x)\sim x^{-θ}$ for $x \ll 1$ and is constant for large $x$. The scaling form implies a fractal distribution of persistent sites with power-law spatial correlations. We study the scaling numerically for Glauber-Ising model at dimension $d = 1$ to 4 and extend the study to the diffusion problem. Our finite size scaling ans…
We reconsider the problem of local persistence in directed site percolation. We present improved estimates of the persistence exponent in all dimensions from 1+1 to 7+1, obtained by new algorithms and by improved implementations of existing ones. We verify the strong corrections to scaling for 2+1 and 3+1 dimensions found in previous analyses, but we show that scaling is much better satisfied for very large and very small dimensions. For d > 4 (d is the spatial dimension), the persistence exponent depends non-trivially on d, in qualitative agreement with the non-universal values calculated rec…
We present simulations of the 1-dimensional Oslo rice pile model in which the critical height at each site is randomly reset after each toppling. We use the fact that the stationary state of this sandpile model is hyperuniform to reach system of sizes $> 10^7$. Most previous simulations were seriously flawed by important finite size corrections. We find that all critical exponents have values consistent with simple rationals: $ν=4/3$ for the correlation length exponent, $D =9/4$ for the fractal dimension of avalanche clusters, and $z=10/7 $ for the dynamical exponent. In addition we relate the…
In this work a three-states cellular automaton proposed to describe part of a biological immune system is revisited. We obtain the dynamic critical exponent $z$ of the model by means of a recent technique that mixes different initial conditions. Moreover, by using two distinct approaches, we have also calculated the global persistence exponent $θ_{g}$, related to the probability that the order parameter of the model does not change its sign up to time $t$ [$P(t)\propto t^{-θ_{g}}$].
We analytically study the effect of a uniform shear flow on the persistence properties of coarsening systems. The study is carried out within the anisotropic Ohta-Jasnow-Kawasaki (OJK) approximation for a system with nonconserved scalar order parameter. We find that the persistence exponent theta has a non-trivial value: theta = 0.5034... in space dimension d=3, and theta = 0.2406... for d=2, the latter being exactly twice the value found for the unsheared system in d=1. We also find that the autocorrelation exponent lambda is affected by shear in d=3 but not in d=2.
We consider a particle diffusing in the y-direction, dy/dt=η(t), subject to a transverse shear flow in the x-direction, dx/dt=f(y), where x \ge 0 and x=0 is an absorbing boundary. We treat the class of models defined by f(y) = \pm v_{\pm}(\pm y)^αwhere the upper (lower) sign refers to y>0 (y<0). We show that the particle survives with probability Q(t) \sim t^{-θ} with θ= 1/4, independent of α, if v_{+}=v_{-}. If v_{+} \ne v_{-}, however, we show that θdepends on both αand the ratio v_{+}/v_{-}, and we determine this dependence.
The scaling behaviour of the persistence probability in the critical dynamics is investigated with both the heat-bath and the Metropolis algorithm for the two-dimensional Ising model and Potts model. Special attention is drawn to the dependence on the initial magnetization. The global persistence exponent is measured. Universality is confirmed.
With dynamic Monte Carlo simulations, we investigate the continuous phase transition in the three-dimensional three-state random-bond Potts model. We propose a useful technique to deal with the strong corrections to the dynamic scaling form. The critical point, static exponents $β$ and $ν$, and dynamic exponent $z$ are accurately determined. Particularly, the results support that the exponent $ν$ satisfies the lower bound $ν\geqslant 2/d$.
We establish an exact formula relating the survival probability for certain Lévy flights (viz. asymmetric $α$-stable processes where $α= 1/2$) with the survival probability for the order statistics of the running maxima of two independent Brownian particles. This formula allows us to show that the persistence exponent $δ$ in the latter, non Markovian case is simply related to the persistence exponent $θ$ in the former, Markovian case via: $δ=θ/2$. Thus, our formula reveals a link between two recently explored families of anomalous exponents: one exhibiting continuous deviations from Sparre-And…
We define a block persistence probability $p_l(t)$ as the probability that the order parameter integrated on a block of linear size $l$ has never changed sign since the initial time in a phase ordering process at finite temperature T<T_c. We argue that p_l(t)\sim l^{-zθ_0}f(t/l^z) in the scaling limit of large blocks, where θ_0 is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent θfor large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large n model, and generically it can be used to determine easily θ…
We study the relaxation for growing interfaces in quenched disordered media. We use a directed percolation depinning model introduced by Tang and Leschhorn for 1+1-dimensions. We define the two-time autocorrelation function of the interface height C(t',t) and its Fourier transform. These functions depend on the difference of times t-t' for long enough times, this is the steady-state regime. We find a two-step relaxation decay in this regime. The long time tail can be fitted by a stretched exponential relaxation function. The relaxation time is proportional to the characteristic distance of the…
We investigate the persistence properties of critical d-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d'-dimensional manifold. The persistence probability P(t) shows three distinct long-time decays depending on the value of the parameter ζ= (D-2+η)/z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d-d' and of the critical expon…
We have investigated the random walk problem in a finite system and studied the crossover induced in the the persistence probability scales by the system size.Analytical and numerical work show that the scaling function is an exponentially decaying function.The particle here is trapped with in a box of size $L$ . We have also considered the problem when the particle in trapped in a potential. Direct calculation and numerical result show that the scaling function here also an exponentially decaying function. We also present numerical works on harmonically trapped randomly accelerated particle a…
We investigate both the local and global persistence behaviour in ANNNI (axial next-nearest neighour Ising) model. We find that when the ratio $κ$ of the second neighbour interaction to the first neighbour interaction is less than 1, $P(t)$, the probability of a spin to remain in its original state upto time $t$ shows a stretched exponential decay. For $κ> 1$, $P(t)$ has a algebraic decay but the exponent is different from that of the nearest neighbour Ising model. The global persistence behaviour shows similar features. We also conduct some deeper investigations in the dynamics of the ANNNI m…
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are investigated here numerically from the point of view of the underlying spin system. The dynamical persistency exponent $Θ$ and the exponent $lambda$ characterising the two-time autocorrelation function of the total magnetization under non-equilibrium conditions are reported. It is found that the PC transition has strong effect: the process becomes non-Markovi…
We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the `partial survival' probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculate discretely sampled persistence exponents of GS…
The spatial distribution of unvisited/persistent sites in $d=1$ $A+A\to\emptyset$ model is studied numerically. Over length scales smaller than a cut-off $ξ(t)\sim t^{z}$, the set of unvisited sites is found to be a fractal. The fractal dimension $d_{f}$, dynamical exponent $z$ and persistence exponent $θ$ are related through $z(1-d_{f})=θ$. The observed values of $d_{f}$ and $z$ are found to be sensitive to the initial density of particles. We argue that this may be due to the existence of two competing length scales, and discuss the possibility of a crossover at late times.
We study the statistical properties of the sum $S_t=\int_{0}^{t}dt' σ_{t'}$, that is the difference of time spent positive or negative by the spin $σ_{t}$, located at a given site of a $D$-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of $S_{t}$ and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ($T>T_{c}$), criticality ($T=T_c$), and low temperature ($T<T_{c}$). We discuss in particular the question of the temperature dependence of the pers…
The local persistence probability P_l(t) that a site never becomes active up to time t, and the global persistence probability P_g(t) that the deviation of the global density from its mean value rho(t)-<ρ(t)> does not change its sign up to time t are studied in a one-dimensional directed percolation process by Monte Carlo simulations. At criticality, starting from random initial conditions, both P_l(t) and P_g(t) decay algebraically with exponents theta_l ~ theta_g ~ 1.50(2), which is in contrast to previously known cases where theta_g < theta_l. The exponents are found to be independent of th…
In this paper, we present the detailed calculation of the persistence exponent $θ$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $θ$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $θ$ for non-Gaussian processes, by making a strong connection between the problem of pe…
We revisit the problem of local persistence in directed percolation, reporting improved estimates of the persistence exponent in 1+1 dimensions, discovering strong corrections to scaling in higher dimensions, and investigating the mean field limit. Moreover, we introduce a graded persistence probability that a site does not flip more than n times and demonstrate how local persistence can be studied in seed simulations. Finally, the problem of spatial (as opposed to temporal) persistence is investigated.
We investigate the critical dynamics of the time-dependent Landau-Ginzburg model with non conserved n-component order parameter (Model A) in the presence of long-range correlated quenched impurities. We use a special kind of long-range correlations, previously introduced by Weinrib and Halperin. Using a double expansion in εand δwe calculate the critical exponent z up to second order on the small parameters. We show that the quenched impurities of this kind affect the critical dynamics already in first order of εand δ, leading to a relevant correction for the mean field value of the exponent z
The probabilities $P_\pm(t_0,t)$ that a growing Kardar-Parisi-Zhang interface remains above or below the mean height in the time interval $(t_0, t)$ are shown numerically to decay as $P_\pm \sim (t_0/t)^{θ_\pm}$ with $θ_+ = 1.18 \pm 0.08$ and $θ_- = 1.64 \pm 0.08$. Bounds on $θ_\pm$ are derived from the height autocorrelation function under the assumption of Gaussian statistics. The autocorrelation exponent $\bar λ$ for a $d$--dimensional interface with roughness and dynamic exponents $β$ and $z$ is conjectured to be $\bar λ= β+ d/z$. For a recently proposed discretization of the KPZ equation …
We study the statistics of simulated earthquakes in a quasistatic model of two parallel heterogeneous faults within a slowly driven elastic tectonic plate. The probability that one fault remains dormant while the other is active for a time Dt following the previous activity shift is proportional to the inverse of Dt to the power 1+x, a result that is robust in the presence of annealed noise and strength weakening. A mean field theory accounts for the observed dependence of the persistence exponent x as a function of heterogeneity and distance between faults. These results continue to hold if t…
We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the $2d$ ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value $θ$ that depends upon the initial condition. More precisely, we find that $θ$ takes one universal value $0.199(2)$ for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value $0.033(1)$ for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with …
A new ''static'' renormalization group approach to stochastic models of fluctuating surfaces with spatially quenched noise is proposed in which only time-independent quantities are involved. As examples, quenched versions of the Kardar-Parisi-Zhang model and its Pavlik's modification, the Hwa-Kardar model of self-organized criticality, and Pastor-Satorras-Rothman model of landscape erosion are studied. It is shown that the upper critical dimension in the quenched models is shifted by two units upwards in comparison to their counterparts with white in-time noise. Possible scaling regimes associ…
We consider classical spin models of two- and three-dimensional spins with continuous symmetry and investigate the effect of a symmetry-breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean-field regime. We show, by perturbative calculations and numerical simulations, that although the continuous symmetry of the magnetization is lost due to disorder, the system still magnetizes in specific directions, albeit with a lower value as compared to the case without disorder. The critical temperature, at which the system starts magnetizing, as well as the ma…
