Non-triviality of a discrete Bak-Sneppen evolution model

Abstract

Consider the following evolution model, proposed in \cite{BS} by Bak and Sneppen. Put $N$ vertices on a circle, spaced evenly. Each vertex represents a certain species. We associate with each vertex a random variable, representing the `state' or `fitness' of the species, with values in $[0,1]$. The dynamics proceeds as follows. Every discrete time step, we choose the vertex with minimal fitness, and assign to this vertex, and to its two neighbours, three new independent fitnesses with a uniform distribution on $[0,1]$. A conjecture of physicists, based on simulations, is that in the stationary regime, the one-dimensional marginal distributions of the fitnesses converges, when $N \to \infty$, to a uniform distribution on $(f,1)$, for some threshold $f<1$. In this paper we consider a discrete version of this model, proposed in \cite{BK}. In this discrete version, the fitness of a vertex can be either 0 or 1. The system evolves according to the following rules. Each discrete time step, we choose an arbitrary vertex with fitness 0. If all the vertices have fitness 1, then we choose an arbitrary vertex with fitness 1. Then we update the fitnesses of this vertex and of its two neighbours by three new independent fitnesses, taking value 0 with probability $0<q<1$, and 1 with probability $p=1-q$. We show that if $q$ is close enough to one, then the mean average fitness in the stationary regime is bounded away from 1, uniformly in the number of vertices. This is a small step in the direction of the conjecture mentioned above, and also settles a conjecture mentioned in \cite{BK}. Our proof is based on a reduction to a continuous time particle system.