Block persistence

We define a block persistence probability p(l)(t) as the probability that the order parameter integrated on a block of linear size l has never changed sign since the initial time in a phase-ordering process at finite temperature T < T-c. We argue that p(l)(t) similar to l(-z theta O) f(t/l(z)) in the scaling limit of large blocks, where z is the growth exponent (L(t) similar to t(1/z)), theta(0) is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent theta for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large-n model, and generically it can be used to determine easily theta(0) from simulations of coarsening models. We also argue that theta(0) and the scaling function do not depend on temperature, leading to a definition of theta at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. These ideas are applied to the study of persistence for conserved models. We illustrate our discussions by extensive numerical results. We also comment on the relation between this method and an alternative definition of theta at finite temperature recently introduced by Derrida [Phys. Rev. E55, 3705 (1997)].