Dynamic scaling in the spatial distribution of persistent sites

The spatial distribution of persistent (unvisited) sites in one dimensional $A+A\to\emptyset$ model is studied. The `empty interval distribution' $n(k,t)$, which is the probability that two consecutive persistent sites are separated by distance $k$ at time $t$ is investigated in detail. It is found that at late times this distribution has the dynamical scaling form $n(k,t)\sim t^{-\theta}k^{-\tau}f(k/t^{z})$. The new exponents $\tau$ and $z$ change with the initial particle density $n_{0}$, and are related to the persistence exponent $\theta$ through the scaling relation $z(2-\tau)=\theta$. We show by rigorous analytic arguments that for all $n_{0}$, $1< \tau< 2$, which is confirmed by numerical results.