In standard bootstrap percolation, a subset A of the grid [n] is initially infected. A new site is then infected if at least two of its neighbours are infected, and an infected site stays infected forever. The set A is said to percolate if eventually the entire grid is infected. A percolating set is said to be minimal if none of its subsets percolate. Answering a question of Bollobás, we show t...

We provide a derivation of the percolation threshold for a two-dimensional triangular lattice. We then proceed to model an ensemble of skewed square lattices with randomly punched out squares, which is topologically equivalent to triangular lattice percolation. We investigate the behavior of the magnitude of the Casimir force between the percolating plate and the solid plate. We find evidence w...

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

Awide variety ofmethods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrappi...

A wide variety of methods have been used to compute percolation thresholds. In lattice percolation, the most powerful of these methods consists of microcanonical simulations using the union-find algorithm to efficiently determine the connected clusters, and (in two dimensions) using exact values from conformal field theory for the probability, at the phase transition, that various kinds of wrap...

I use a previously introduced mapping between the continuum percolation model and the Potts fluid to derive a mean field theory of continuum percolation systems. This is done by introducing a new variational principle, the basis of which has to be taken, for now, as heuristic. The critical exponents obtained are β = 1, γ = 1 and ν = 0.5, which are identical with the mean field exponents of latt...

Let (Xn, rn)n≥1 be a marked Poisson process with intensity λ in Rd, d ≥ 2. The marks (rn) are radii of closed Euclidean balls centered at the points (Xn). Two points Xi and Xj of the Poisson process X are adjacent, Xi ∼ Xj , if D(Xi, ri) ∩ D(Xj , rj) 6= ∅, where D(x,R) = {y ∈ Rd : ||x− y||2 ≤ R}. We say that x, y ∈ Rd are connected, x↔ y, if there are Xi1 , . . . , Xil ∈ X such that x ∈ D(Xi1 ,...

The goal of this set of lectures is to combine two seemingly unrelated topics: • The study of Boolean functions, a field particularly active in computer science • Some models in statistical physics, mostly percolation The link between these two fields can be loosely explained as follows: a percolation configuration is built out of a collection of i.i.d " bits " which determines whether the corr...

Blogs facilitate online debates and discussions for millions of people around the world. Identifying the most popular and prevailing topics discussed in the Blogosphere is a crucial task. This paper describes our novel approach to the quantification of the level of topic propagation in the Blogosphere. It tries to answer one key question: How many people should know about a subject before it be...

Bootstrap percolation, one of the simplest cellular automata, can be seen as a model of the spread of infection. In r-neighbour bootstrap percolation on a graph G we assign a state, infected or healthy, to every vertex of G and then update these states in successive rounds, according to the following simple local update rule: infected vertices of G remain infected forever and a healthy vertex b...