Phase Transitions for the Groeth Rate of Linear Stochastic Evolutions
  We consider a simple discrete-time Markov chain with values in $[0,\infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented 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 $d \ge 3$ and the Markov chain 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 Markov chain is "random enough", then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization. 
Nobuo Yoshida