Annals of Probability

Limit behavior of the Bak--Sneppen evolution model

Ronald Meester and Dmitri Znamenski

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Abstract

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.

Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 1986-2002.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1068646375

Digital Object Identifier
doi:10.1214/aop/1068646375

Mathematical Reviews number (MathSciNet)
MR2016609

Zentralblatt MATH identifier
1044.60095

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B26: Phase transitions (general) 92D15: Problems related to evolution

Keywords
Bak--Sneppen species fitness evolution interacting particle system self-organized criticality coupling stationary distribution phase transition graph

Citation

Meester, Ronald; Znamenski, Dmitri. Limit behavior of the Bak--Sneppen evolution model. Ann. Probab. 31 (2003), no. 4, 1986--2002. doi:10.1214/aop/1068646375. https://projecteuclid.org/euclid.aop/1068646375


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References

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