  We study the distribution of persistent sites (sites unvisited by particles
$A$) in one dimensional $A+A\to\emptyset$ reaction-diffusion model. We define
the {\it 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 $s\sim t^{1/2}$. We show
under the IIA that the scaling function $f(x)\sim x^{-\tau}$ as $x\to 0$ and
decays exponentially at large $x$. The exponent $\tau$ is related to the
persistence exponent $\theta$ 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 $r\ll s$, $C(r,t)\sim r^{-\alpha}$ where $\alpha=2-\tau$, in agreement
with our earlier numerical results.
