Dynamic crossover in the persistence probability of manifolds at criticality

Abstract

We investigate the persistence properties of criticald-dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., themagnetization in the Ising model), focusing on the dynamics of the global order parameter of ad′-dimensional manifold. The persistence probabilityPc(t) shows three distinct long-time decays depending on the value of the parameterζ = (D − 2 + η)/z which also controls the relaxation of the persistence probability in the case of adisordered initial state (vanishing order parameter) as a function of the codimensionD = d − d′ and of the criticalexponents η and z. We find that the asymptotic behavior ofPc(t) is exponentialfor ζ > 1, stretchedexponential for 0 ≤ ζ ≤ 1,and algebraic for ζ < 0. Whereas the exponential and stretched exponential relaxations are not affected by theinitial value of the order parameter, we predict and observe a crossover between twodifferent power-law decays when the algebraic relaxation occurs, as in the cased′ = d of the global order parameter. We confirm via Monte Carlo simulations our analyticalpredictions by studying the magnetization of a line and of a plane of the two- andthree-dimensional Ising models, respectively, with Glauber dynamics. The measured exponentsof the ultimate algebraic decays are in a rather good agreement with our analyticalpredictions for the Ising universality class. In spite of this agreement, the expected scalingbehavior of the persistence probability as a function of time and of the initial value of theorder parameter remains problematic. In this context, the non-equilibrium dynamics of theO(n) model in the limit and its subtle connection with the spherical model are also discussed in detail. Inparticular, we show that the correlation functions of the components of the orderparameter which are respectively parallel and transverse to its average value within the model correspond to the correlation functions of the local and global order parameters ofthe spherical model.