Consider a (discrete time) spatial branching process on the site space of nonnegative integers. In each time step, at each site, particles independently produce offspring according to an offspring distribution that is drawn at random for this specific site (but does not depend on time). Each offspring particle, with a probability that is sampled at random for this specific site, makes a jump to the right or otherwise simply stays where it is. Both the offspring law and the jump probability are drawn in an i.i.d. fashion and form the random environment \(\omega\) for the process. Denote by \(m\) the (random) mean of the generic offspring law, by \(M\) its essential supremum, and by \(h\) the generic jump probability. Finally denote by \(\Lambda\) the essential supremum of \(m(1-h)\).
Starting with one particle at the origin, the authors show that there is local survival (some site is populated forever) for almost all environments if \(\Lambda>1\). Otherwise there is no local survival for almost all environments. Furthermore, the authors show a 0-1 law for global survival, the criterion being that either \(\Lambda>1\) or the \(\mathbf{E}[\log(mh/(1-m(1-h)))]>0\). Finally, the authors show exponential growth of the population on the event of non-extinction and prove convergence to a deterministic spatial profile if the population is properly rescaled in space and size.