  We describe a universal transition mechanism characterizing the passage to an
annealed behavior and to a regime where the fluctuations about this behavior
are Gaussian, for the long time asymptotics of the empirical average of the
expected value of the number of random walks which branch and annihilate on
${\mathbb Z}^d$, with stationary random rates. The random walks are
independent, continuous time rate $2d\kappa$, simple, symmetric, with $\kappa
\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$,
and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates
$w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe
the annealed-Gaussian transition mechanism under mild conditions on the rates.
Indeed, we introduce the exponents
$F_\theta(t):=\frac{H_1((1+\theta)t)-(1+\theta)H_1(t)}{\theta}$, and assume
that $\frac{F_{2\theta}(t)-F_\theta(t)}{\theta\log(\kappa t+e)}\to\infty$ for
$|\theta|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $<m(0,t)>$
denotes the average of the expected value of the number of particles $m(0,t,w)$
at time $t$ and an environment of rates $w$, given that initially there was
only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of
side $L(t)$ has different behaviors: if $ L(t)\ge e^{\frac{1}{d}
F_\epsilon(t)}$ for some $\epsilon >0$ and large enough $t$, a law of large
numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_\epsilon (2t)}$ for some
$\epsilon>0$ and large enough $t$, a CLT is satisfied. These statements are
violated if the reversed inequalities are satisfied for some negative
$\epsilon$. Applications to potentials with Weibull, Frechet and double
exponential tails are given.
