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-c(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 eta and z. We find that the asymptotic behavior of P-c(t) is exponential for zeta > 1, stretched exponential for 0 =. = 1, and algebraic for zeta infinity and its subtle connection with the spherical model are also discussed in detail. In particular, show that the correlation functions of the components of the order parameter which are respectively parallel and transverse to its average value within the O(n -> infinity) model correspond to the correlation functions of the local and global order parameters of the spherical model