Two-species diffusion-annihilation process on the fully-connected lattice: probability distributions and extreme value statistics

International audienceWe study the two-species diffusion-annihilation process, A + B → Ø, on the fully-connected lattice. Probability distributions for the number of particles and the reaction time are obtained for a finite-size system using a master equation approach. Mean values and variances are deduced from generating functions. When the reaction is far from complete, i.e. for a large number of particles of each species, mean-field theory is exact and the fluctuations are Gaussian. In the scaling limit the reaction time displays extreme-value statistics in the vicinity of the absorbing states. A generalized Gumbel distribution is obtained for unequal initial densities, ρ A > ρ B. For equal or almost equal initial densities, ρ A ≃ ρ B , the fluctuations of the reaction time near the absorbing state are governed by a probability density involving derivatives of ϑ 4 , the Jacobi theta function