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We study branching random walks in random environment on the $d$-dimensionalsquare lattice, $d \geq 1$. In this model, the environment has finite rangedependence, and the population size cannot decrease. We prove limit theorems(laws of large numbers) for the set of lattice sites which are visited up to alarge time as well as for the local size of the population. The limiting shapeof this set is compact and convex, though the local size is given by a concavegrowth exponent. Also, we obtain the law of large numbers for the logarithm ofthe total number of particles in the process.

Shape and local growth for multidimensional branching random walks in random environment