Limit behavior of the Bak--Sneppen evolution model

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.