Dynamic crossover in the persistence probability of manifolds at criticality

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 zeta = (D-2+eta)/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 exponents z and eta. We find that the asymptotic behavior of P(t) is exponential for zeta > 1, stretched exponential for 0 <= zeta <= 1, and algebraic for zeta < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d=d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising model, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O(n) model in the limit n->infty and its subtle connection with the spherical model is also discussed in detail.

Slow relaxation, dynamic transitions and extreme value statistics in disordered systems

We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape.