Wilson renormalization of a reaction-diffusion process

Abstract

Healthy and sick individuals (A and B particles) diffuse independently with diffusion constants D_A and D_B. Sick individuals upon encounter infect healthy ones (at rate k), but may also spontaneously recover (at rate 1/τ). The propagation of the epidemic therefore couples to the fluctuations in the total population density. Global extinction occurs below a critical value ρ_{c} of the spatially averaged total density. The epidemic evolves as the diffusion--reaction--decay process A + B --> 2B, B --> A , for which we write down the field theory. The stationary state properties of this theory when D_A=D_B were obtained by Kree et al. The critical behavior for D_A<D_B is governed by a new fixed point. We calculate the critical exponents of the stationary state in an $\eps$ expansion, carried out by Wilson renormalization, below the critical dimension d_{c}=4. We then go on to to obtain the critical initial time behavior at the extinction threshold, both for D_A=D_B and D_A<D_B. There is nonuniversal dependence on the initial particle distribution. The case D_A>D_B remains unsolved.