Diffusion and Multiplication in Random Media

Abstract : We investigate the evolution of a population of non-interacting particles which undergo diffusion and multiplication. Diffusion is assumed to be homogeneous, while multiplication proceeds with different rates reflecting the distribution of nutrients. We focus on the situation where the distribution of nutrients is a stationary quenched random variable, and show that the population exhibits a super-exponential growth whenever the nutrient distribution is unbounded. We elucidate a huge difference between the $average$ and $typical$ asymptotic growths and emphasize the role played by the spatial correlations in the nutrient distribution.