Non-triviality of a discrete Bak-Sneppen evolution model

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

Extremal point of infinite clusters in stationary percolation

It is well known that in stationary percolation an innite component cannot have a nite number of extremal points in a certain direction In this note we investigate whether or not an innite cluster can have innitely many extremal points in a certain direction To make this question at all interesting it is necessary and natural to simultaneously ask for an innite path in the opposite direction It turns out that the answer depend on the dimension of the model and on the question whether or not the model has socalled nite rang

A note on percolation in cocycle measures

We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a very specific form and direction. In concrete situations, this leads to a quick decision whether or not a certain cocycle measure percolates

A simple proof of the exponential convergence of the modified Jacobi-Perron algorithm

It has recently been shown in Ito et al that the modied JacobiPerron algorithm is strongly convergent in the sense of Brentjes almost everywhere with exponential rate Their proof relies on very complicated computations In this paper we will show that the original paper of Podsypanin on the modied Jacobi-Perron algorithm almost contains a proof of this convergence with exponential rate The only ingredients missing in that paper are some ergodic-theoretical facts about the transformation generating the approximations This leads to a very simple proof of the beforementioned exponential convergence in the modied JacobiPerron algorith

On a long range particle system with unbounded flip rates

We consider an interacting particle system on f0; 1g Z with non-local, unbounded ip rates. Zeroes ip to one at a rate that depends on the number of ones to the right until we see a zero (the ip rate equals times one plus this number). The ip rate of the ones equals . We give motivation for models like this in general, and this one in particular. The system turns out not to be Feller, and we construct it using monotonicity. We show thatfor < the system has a unique non-trivial stationary distribution, which is ergodic, stationary, and has a density ofones of . For the limit is degenerate at f1g Z . Our main tool is an explicit formula for the density of ones at any given moment

Stability and weakly convergent approximations of queuing systems on a circle

We rst consider a nongreedy queueing system on a circle We present a new and very simple proof of the stability of this system un der the appropriate condition based on the average travel times between customers Next we show that the same nongreedy system with a restricted number of customers converges weakly to this system when the restricted number goes to innity Finally we consider a polling network with nitely many service stations in which the server has a greedy service strategy Under the appropriate condition we give a new simple proof of the stability of this syste

Entropy for random group actions

We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a Z d action. We show that the classical denition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when d 2. We propose a denition of the entropy for random group actions which agrees with the classical denition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This denition is based on the bre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases

A field test of six types of live-trap for African rodents

Six live-trap types were tested in a grassy vlei near Pretoria in an attempt to determine their success in trapping Rhabdomys pumilio, Praomys (Mastomys) natalensis, and Otomys irroratus. One trap of each type was set at each of fourteen trapping stations. The effect of trap position on captures was effectively ruled out by changing the arrangement of the traps each week. The frequency of capture of different age and sex classes of the three species is compared