Dynamic scaling in the spatial distribution of persistent sites

Abstract

The spatial distribution of persistent (unvisited) sites in one dimensional $A+A\to \varnothing$ 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 $n(k,t)\sim t^{-\hat{I}¸}{k}^{-\ddot{I}„}f(k/t^z)$. The new exponents $\ddot{I}„$ and $z$ change with the initial particle density $n_0$, and are related to the persistence exponent $\hat{I}¸$ through the scaling relation $z(2-\ddot{I}„)=\hat{I}¸$. We show by rigorous analytic arguments that for all $n_0$, $1<\ddot{I}„<2$, which is confirmed by numerical results.

4 pages, REVTEX, 4 postscript figures