Dynamic scaling in the spatial distribution of persistent sites

The spatial distribution of persistent (unvisited) sites in one dimensional $A+A\to \mathrm{\varnothing }$ 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 $n(k,t)\sim t^{-\theta}k^{-\tau}f(k/t^{z})$\mathrm{.}Thenewexponents$\tau$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.