Spatial distribution of persistent sites

We study the distribution of persistent sites (sites unvisited by particles A) in the one-dimensional A+A --> empty set reaction-diffusion model. We define the empty intervals as the separations between adjacent persistent sites, and study their size distribution n(k, t) as a function of interval length k and time t. The decay of persistence is the process of irreversible coalescence of these empty intervals, which we study analytically under the independent interval approximation (IIA). Physical considerations suggest that the asymptotic solution is given by the dynamic scaling form n(k, t) = s(-2) f(k/s) with the average interval size s similar to t(1/2). We show under the IIA that the scaling function f(x) similar to x(-tau) as x --> 0 and decays exponentially at large x. The exponent tau is related to the persistence exponent a through the scaling relation tau = 2(1 - theta). We compare these predictions with the results of numerical simulations. We determine the two-point correlation function C(r, t) under the IIA. We find that for r much less than s, C(r, t) similar to r(-alpha) where alpha = 2 - tau, in agreement with our earlier numerical results.