Phase Transition Asymptotics for Random Walks on a Stationary Random Potential

Abstract. 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 Z, with stationary random rates. The random walks are independent, continuous time rate 2dκ, simple, symmetric, with κ ≥ 0. A random walk at x ∈ Z, 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 Fθ(t) := H1((1+θ)t)−(1+θ)H1(t) θ , and assume that F2θ(t)−Fθ(t) θ log(κt+e) → ∞ for |θ| > 0 small enough, where H1(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) ≥ 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 universality 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. Our results indicate the presence of a mean field type phase transition mechanism, generalizing the law of large numbers and the CLT proved in [BBM(2005)] and in [BMR(2005)].