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October 2003 Limit behavior of the Bak--Sneppen evolution model
Ronald Meester, Dmitri Znamenski
Ann. Probab. 31(4): 1986-2002 (October 2003). DOI: 10.1214/aop/1068646375


One of the key problems related to the Bak--Sneppen evolution model on the circle is computing the limit distribution of the fitness at a fixed observation vertex in the stationary regime as the size of the system tends to infinity. Some simulations have suggested that this limit distribution is uniform on $(f,1)$ for some $f\sim2/3$. In this article, we prove that the mean of the fitness in the stationary regime is bounded away from 1, uniformly in the size of the system, thereby establishing the nontriviality of the limit behavior. The Bak--Sneppen dynamics can easily be defined on any finite connected graph. We also present a generalization of the phase-transition result in the context of an increasing sequence of such graphs. This generalization covers the multidimentional Bak--Sneppen model as well as the Bak--Sneppen model on a tree. Our proofs are based on a "self-similar'' graphical representation of the avalanches.


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Ronald Meester. Dmitri Znamenski. "Limit behavior of the Bak--Sneppen evolution model." Ann. Probab. 31 (4) 1986 - 2002, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1044.60095
MathSciNet: MR2016609
Digital Object Identifier: 10.1214/aop/1068646375

Primary: 60K35 , 82B26 , 92D15

Keywords: Bak--Sneppen , coupling , Evolution , fitness , graph , Interacting particle system , phase transition , Self-organized criticality , species , stationary distribution

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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