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 d efine the empty intervals as the separations between adjacent persistent si tes, and study their size distribution n(k, t) as a function of interval le ngth 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 u nder the IIA that the scaling function f(x) similar to x(-tau) as x --> 0 a nd decays exponentially at large x. The exponent tau is related to the pers istence exponent a through the scaling relation tau = 2(1 - theta). We comp are these predictions with the results of numerical simulations. We determi ne the two-point correlation function C(r, t) under the IIA. We find that f or r much less than s, C(r, t) similar to r(-alpha) where alpha = 2 - tau, in agreement with our earlier numerical results.