J ul 2 00 7 TRANSITION ASYMPTOTICS FOR REACTION - DIFFUSION IN RANDOM MEDIA

We describe a universal transition mechanism between annealed and quenched regimes in the context of reaction-diffusion in random media. We study the total population size for random walks which branch and annihilate on Z d , with time-independent random rates. The random walks are independent , continuous time, rate 2dκ, simple, symmetric, with κ ≥ 0. A random walk at x ∈ 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 θ (t) := H 1 ((1+θ)t)−(1+θ)H 1 (t) θ , and assume that F 2θ (t)−F θ (t) θ log(κt+e) → ∞ for |θ| > 0 small enough, where H1(t) := logm(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) ≥ e 1 d Fǫ(t) for some ǫ > 0 and large enough t, a law of large numbers is satisfied; if L(t) ≥ e 1 d Fǫ(2t) for some ǫ > 0 and large enough t, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative ǫ. As corollaries, we obtain more explicit statements under regularity conditions on the tails of the random rates, including examples in the four univer-sality classes defined in [HKM(2005)]: potentials which are unbounded of Weibull type, of double exponential type, almost bounded, and bounded of Fréchet type. For them we also derive sharper results in the non-annealed regime.