Eur. Phys. J. B 7, 111-127 (1999)
https://doi.org/10.1007/s100510050594

Block persistence

Laboratoire de Physique Quantique (CNRS -UMR C5626 ) , Université Paul Sabatier, 31062 Toulouse Cedex, France

Received: 25 February 1998
Revised: 24 July 1998
Accepted: 27 July 1998
Published online: 15 January 1999

Abstract

We define a block persistence probability 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 . We argue that in the scaling limit of large blocks, where z is the growth exponent (), is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent θ 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 from simulations of coarsening models. We also argue that and the scaling function do not depend on temperature, leading to a definition of θ 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 θ at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)].