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

We study the two-species diffusion-annihilation process, $A+B\rightarrow$ \O, 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, $\rho_A>\rho_B$. For equal or almost equal initial densities, $\rho_A\simeq\rho_B$, the fluctuations of the reaction time near the absorbing state are governed by a probability density involving derivatives of $\vartheta_4$, the Jacobi theta function.