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

The distribution $n(k,t)\mathrm{}$ 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 $\theta \mathrm{}\left(q\right)\mathrm{}$ is the persistence exponent which describes the fraction $P\left(t\right)\mathrm{}\sim t^{-\theta }$ 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.