Abstract
We study a discrete-time multitype branching random walk on a finite space with finite set of types. Particles move in space according to a Markov chain whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman–Kac formula. We choose Weibull-type distributions with parameter \(1/\rho _{ij}\) for the upper tail of the mean number of \(j\) type particles produced by an \(i\) type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.
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