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 the global persistence exponent that characterizes this decay is while for d<4 its value is increased to first order in . Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to to first order in . The global persistence exponent is found to be a new and independent exponent. Finally we compare our results with existing simulations.
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 the global persistence exponent that characterizes this decay is while for d<4 its value is increased to first order in . Combining a method developed by Majumdar and Sire with renormalization group techniques we compute the correction to to first order in . The global persistence exponent is found to be a new and independent exponent. Finally we compare our results with existing simulations.
The persistence exponent $\ensuremath{\theta}$ for the global order parameter M(t) of a system quenched from the disordered phase to its critical point describes the probability, $p(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}$, that M(t) does not change sign in the time interval t following the quench. We calculate $\ensuremath{\theta}$ to $O({\ensuremath{\epsilon}}^{2})$ for model A of Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)] (and to order $\ensuremath{\epsilon}$ for model C) and show that at this order M(t) is a non-Markov process. Consequently, to our knowledge, $\ensuremath{\theta}$ is 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. Rev. Lett. 77, 1420 (1996)].
We investigate global persistence properties for the non-equilibrium critical dynamics of the randomly diluted Ising model. The disorder-averaged persistence probability c ( t ) of the global magnetization is found to decay algebraically with an exponent θ c that we compute analytically in a dimensional expansion in d = 4 − . Corrections to Markov process are found to occur already at one loop order and θ c is thus a novel exponent characterizing this disordered critical point. Our result is thoroughly compared with Monte Carlo simulations in d = 3, which also include a measurement of the initial slip exponent. Taking carefully into account corrections to scaling, θ c is found to be universal, independent of the dilution factor p along the critical line at T c ( p ), and in good agreement with our one-loop calculation.
A ``persistence exponent'' $\ensuremath{\theta}$ is defined for nonequilibrium critical phenomena. It describes the probability, $p(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\theta}}$, that the global order parameter has not changed sign in the time interval t following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ limit of the O(n) model, to first order in $\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4\ensuremath{-}d$, and for the 1D Ising model. Numerical results are obtained for the 2D Ising model. We argue that $\ensuremath{\theta}$ is a new independent exponent.
We study the time dependence of the local persistence probability during a nonstationary time evolution in the disordered contact process in d=1, 2, and 3 dimensions. We present a method for calculating the persistence with the strong-disorder renormalization group (SDRG) technique, which we then apply at the critical point analytically for d=1 and numerically for d=2,3. According to the results, the average persistence decays at late times as an inverse power of the logarithm of time, with a universal dimension-dependent generalized exponent. For d=1, the distribution of sample-dependent local persistence is shown to be characterized by a universal limit distribution of effective persistence exponents. Using a phenomenological approach of rare-region effects in the active phase, we obtain a nonuniversal algebraic decay of the average persistence for d=1 and enhanced power laws for d>1. As an exception, for randomly diluted lattices, the algebraic decay remains valid for d>1, which is explained by the contribution of dangling ends. Results on the time dependence of average persistence are confirmed by Monte Carlo simulations. We also prove the equivalence of the persistence with a return probability, a valuable tool for the argumentations.
The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.
We study the persistence probability P(t) that, starting from a random initial condition, the magnetization of a ${d}^{\ensuremath{'}}$-dimensional manifold of a d-dimensional spin system at its critical point does not change sign up to time t. For ${d}^{\ensuremath{'}}>0$ we find three distinct late-time decay forms for P(t): exponential, stretched exponential, and power law, depending on a single parameter $\ensuremath{\zeta}=(D\ensuremath{-}2+\ensuremath{\eta})/z$, where $D=d\ensuremath{-}{d}^{\ensuremath{'}}$ and $\ensuremath{\eta},z$ are standard critical exponents. In particular, we predict that for a line magnetization in the critical d=2 Ising model, P(t) decays as a power law while, for d=3, P(t) decays as a power of t for a plane magnetization but as a stretched exponential for a line magnetization. Numerical results are consistent with these predictions.