No exceptional words for Bernoulli percolation

Exceptional Planes of Percolation

Consider the standard continuous percolation in R 4 , and choose the parameters so that the induced percolation on a xed two dimensional linear subspace is critical. Although two dimensional critical percolation dies, we show that there are exceptional two dimensional linear subspaces, in which percolation occurs.

Exceptional Words Counting

Competition between Growths Governed by Bernoulli Percolation

We study a competition model on Z where the two infections are driven by supercritical Bernoulli percolations with distinct parameters p and q. We prove that, for any q, there exist at most countably many values of p < min {q,− pc} such that coexistence can occur.

Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters “freeze” (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N , the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in [14, 6]. Using volume as a way to measure the size of a cluster – instead of diameter – leads, for large N , to ...

Large deviations for the chemical distance in supercritical Bernoulli percolation

The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm μ depending on the dimension and the percolation parameter, the probability of the event