stochastic growth models in random environments and their phase transition

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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 and the voter model. We study the phase transition for the growth rate of the ``total number of particles" in this framework.The main results are roughly as follows: If d ≧ 3 and the system is ``not too random", then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, d=1,2, or the system is ``random enough", then the growth rate is slower than its expectation.