Blocking and Persistence in the Zero-Temperature Dynamics of Homogeneous and Disordered Ising Models

A “persistence” exponent θ has been extensively used to describe the nonequilibrium dynamics of spin systems following a deep quench: for zero-temperature homogeneous Ising models on the d-dimensional cubic lattice Zd, the fraction p(t) of spins not flipped by time t decays to zero like t−θ(d) for low d; for high d, p(t) may decay to p(∞) > 0, because of “blocking” (but perhaps still like a power). What are the effects of disorder or changes of lattice? We show that these can quite generally lead to blocking (and convergence to a metastable configuration) even for low d, and then present two examples — one disordered and one homogeneous — where p(t) decays exponentially to p(∞). In modelling the nonequilibrium dynamics of spin systems following a deep quench, the following question naturally arises [1, 2, 3, 4, 5, 6, 7, 8]: given a spin system at zero temperature with random starting configuration and evolving according to the usual Glauber dynamics, what is the probability p(t) at time t that a spin has not yet flipped? For the homogeneous ferromagnetic Ising model on Z, this probability has been found to decay at large times as a power law p(t) ∼ t [1, 2, 3] for d < 4. The “persistence” exponent θ(d) is considered to be a new universal exponent governing nonequilibrium dynamics following a deep quench [6]. The persistence problem can be extended to positive temperatures by considering the dynamics of the local order parameter rather than that of single spins [9, 10, 11]. In this paper we confine our attention to dynamics at zero temperature in infinite spin systems. In the usual case of asynchronous updating, a spin is chosen at random (this can be made precise for infinite systems, as in [12]) and then: always flips if the resulting Partially supported by the National Science Foundation under grant DMS-98-02310. Partially supported by the National Science Foundation under grant DMS-98-02153.