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.
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