Localization for Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on d-dimensional lattice. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of “replica overlap”. This extends a result known for the directed polymers in random environment to a large class of models. A new approach, based on the multiplicative Doob’s decomposition, is adopted to overcome the difficulty that the total population may get extinct even at finite time.