Phase Transitions for the Growth Rate of Linear Stochastic Evolutions |
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Authors: | Nobuo Yoshida |
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Institution: | (1) Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan |
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Abstract: | 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. We also discuss the above phase
transition for the dual processes and its connection to the structure of invariant measures for the model with proper normalization.
Supported in part by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 17540112. |
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Keywords: | Phase transition Linear stochastic evolutions Regular growth phase Slow growth phase |
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