Unusual dynamical scaling in the spatial distribution of persistent sites in one-dimensional Potts models

The distribution n(k,t) of the interval sizes k between clusters of persistent sites in the dynamical evolution of the one-dimensional q-state Potts model is studied using a combination of numerical simulations, scaling arguments, and exact analysis. It is shown to have the scaling form n(k,t)=t(-2z)f(k/t(z)), with z=max(1/2,theta), where theta(q) is the persistence exponent which describes the fraction P(t)similar to t(-theta) of sites which have not changed their state up to time t. When theta>1/2, the scaling length t(theta) for the interval-size distribution is larger than the coarsening length scale t(1/2) that characterizes spatial correlations of the Potts variables.