 0893 Nobuo YOSHIDA
 Phase Transitions for the Growth Rate of Linear Stochastic Evolutions
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May 17, 08

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Abstract. We consider a simple discretetime Markov chain with values in
$[0,\8)^{\zd}$. 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.
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