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Rank-driven Markov processes

Grinfeld, Michael; Knight, Philip A.; Wade, Andrew R.

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Authors

Michael Grinfeld

Philip A. Knight



Abstract

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κN on {1,2,…,N}K. Our main results are that, under appropriate conditions on κN, the system exhibits threshold behavior at s∗∈[0,1], where s∗ is a function of κN, and the marginal distribution of a randomly selected element converges to U[s∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.

Citation

Grinfeld, M., Knight, P. A., & Wade, A. R. (2012). Rank-driven Markov processes. Journal of Statistical Physics, 146(2), 378-407. https://doi.org/10.1007/s10955-011-0368-7

Journal Article Type Article
Publication Date Jan 1, 2012
Deposit Date Oct 4, 2012
Publicly Available Date Mar 28, 2024
Journal Journal of Statistical Physics
Print ISSN 0022-4715
Electronic ISSN 1572-9613
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 146
Issue 2
Pages 378-407
DOI https://doi.org/10.1007/s10955-011-0368-7
Keywords Bak–Sneppen evolution model, Self-organized criticality, Markov process on order statistics, Phase transition, Interacting particle system

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Copyright Statement
The original publication is available at www.springerlink.com




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