Block persistence

We define a block persistence probability p(l)(t) as the probability that t he 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 temperat ure T < T-c. We argue that p(l)(t) similar to l(-z theta O) f(t/l(z)) in th e scaling limit of large blocks, where z is the growth exponent (L(t) simil ar 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. Th is scaling is demonstrated at zero temperature for the diffusion equation a nd the large-n model, and generically it can be used to determine easily th eta(0) from simulations of coarsening models. We also argue that theta(0) a nd the scaling function do not depend on temperature, leading to a definiti on of theta at finite temperature, whereas the local persistence probabilit y decays exponentially due to thermal fluctuations. These ideas are applied to the study of persistence for conserved models. We illustrate our discus sions by extensive numerical results. We also comment on the relation betwe en this method and an alternative definition of theta at finite temperature recently introduced by Derrida [Phys. Rev. E55, 3705 (1997)].