NDLI: Rank-Driven Markov Processes

Content Provider	Springer Nature Link
Author	Grinfeld, Michael Knight, Philip A. Wade, Andrew R.
Copyright Year	2011
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 κ $_{ N }$ on {1,2,…,N}$^{ K }$. Our main results are that, under appropriate conditions on κ $_{ N }$, the system exhibits threshold behavior at s $^{∗}$∈[0,1], where s $^{∗}$ is a function of κ $_{ N }$, 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.
Starting Page	378
Ending Page	407
Page Count	30
File Format	PDF
ISSN	00224715
Journal	Journal of Statistical Physics
Volume Number	146
Issue Number	2
e-ISSN	15729613
Language	English
Publisher	Springer US
Publisher Date	2011-10-06
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Bak–Sneppen evolution model Self-organized criticality Markov process on order statistics Phase transition Interacting particle system Quantum Physics Physical Chemistry Theoretical, Mathematical and Computational Physics Statistical Physics, Dynamical Systems and Complexity
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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