Persistent spins in the linear diffusion approximation of phase ordering and zeros of stationary gaussian processes

The fraction r(t) of spins which have never flipped up to time t is studied within a linear diffusion approximation to phase ordering. Numerical simulations show that, even in this simple context, r(t) decays with time like a power-law with a non-trival exponent $\theta$ which depends on the space dimension. The local dynamics at a given point is a special case of a stationary gaussian process of known correlation function and the exponent $\theta$ is shown to be determined by the asymptotic behavior of the probability distribution of intervals between consecutive zero-crossings of this process. An approximate way of computing this distribution is proposed, by taking the lengths of the intervals between successive zero-crossings as independent random variables. The approximation gives values of the exponent $\theta$ in close agreement with the results of simulations.