2008/05/17 by Nobuo Yoshida, Yoshida, Nobuo
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60J37 #60K35 (Primary) #60K37 #82B26 (Secondary) #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60J37 #msc:60K35 #msc:60K37 #msc:82B26
paper · pdf · doi:10.48550/arxiv.0805.2652
21 pages
openalex publication_date 2008/05/17 · arxiv created 2009/06/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a simple discrete-time Markov chain with values in [0,∞)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 ≥ 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.