2005/10/24 by Gérard Ben Arous, Arous, Gerard Ben, Stanislav Molchanov +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #82B41 #82B44 #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.math/0510519
openalex publication_date 2005/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Zd, with stationary random rates. The random walks are independent, continuous time rate 2dκ, simple, symmetric, with κ≥ 0. A random walk at x∈\mathbb Zd, 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):=\fracH1((1+θ)t)-(1+θ)H1(t)θ, and assume that \fracF2θ(t)-Fθ(t)θlog(κt+e)→∞ for |θ|>0 small enough, where H1(t):=log < m(0,t)> and 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) Fε(t) for some ε>0 and large enough t, a law of large numbers is satisfied; if L(t)≥ e(1)/(d) Fε(2t) for some ε>0 and large enough t, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative ε. Applications to potentials with Weibull, Frechet and double exponential tails are given.