Limit behavior of the Bak-Sneppen evolution model

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 ∼ 2/3. 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

Random entropy and recurrence

We show that a cocycle, which is nothing but a generalized random walk with index set ℤd, with bounded step sizes is recurrent whenever its associated random entropy is zero, and transient whenever its associated random entropy is positive. This generalizes a well-known one-dimensional result and implies a Polya type dichotomy for this situation

Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

Covering algorithms, continuum percolation and the geometry of wireless networks

Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius r, have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems. We look at the percolation properties of this generalized model, showing that an unbounded connected component of discs does not exist, almost surely, for small values of the density lambda of the Poisson point process, for any covering algorithm. In general, it turns out not to be true that unbounded connected components arise when lambda is taken sufficiently high. However, we identify some large families of covering algorithms, for which such an unbounded component does arise for large values of lambda. We show how a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of lambda, for any covering algorithm. Finally, we show that a large class of covering algorithms, which arise in many practical applications, can get arbitrarily close to achieving a minimal density of covering discs. We also construct an algorithm that achieves this minimal density

Dimension (in)equalities and H\"older continuous curves in fractal percolation

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial H\"older continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.Comment: 22 pages, 3 figures, accepted for publication in Journal of Theoretical Probabilit