Several Problems in Bernoulli Percolation

Bernoulli percolation is a classical model for random media, which amounts to populate the vertices of a lattice in an independent fashion. For some parameter p, each vertex contains a particle with probability p, and it is left vacant with probability 1-p. A drastic change of behavior can be observed as one lets the parameter vary, at some critical threshold pc: for p < pc, particles form only finite (and tiny) connected components, while for p > pc, a giant (infinite) connected component arises.Percolation theory is widely used in applications, to understand spatial correlations in various systems. An example is the forest fire process of Drossel and Schwabl, introduced in 1992 as a model of excitable media. In such a process, new trees arrive on a lattice with rate 1, while lightning hits the lattice with some rate > 0 (typically very small): when a tree is hit, it burns and fire spreads instantaneously to its neighbors, so that the whole connected component of trees disappears immediately. This model is especially interesting because it provides a paradigmatic example where self-organized criticality arises, a fascinating phenomenon which can be used to explain the emergence of complexity in nature.In this research proposal, we want to improve our mathematical understanding of several key questions related to Bernoulli percolation, both in two dimensions and in higher dimensions. The starting point would be our recent series of works on two-dimensional frozen percolation and forest fires, where we showed in particular the appearance of exceptional scales, near-critical avalanches, and a striking deconcentration phenomenon. For this purpose, a percolation process with impurities played a central role, and we plan to study further this process, with a view to establishing a form of universality in this context. Moreover, our detailed understanding of critical and near-critical percolation in two dimensions allows us to describe carefully the geometry of large critical connected components, and we hope to derive improved estimates on the behavior of random walks in such a random scenery. Finally, we plan to consider questions related to “patterns” which can be observed in a percolation configuration, so as to extend in various directions some important results obtained recently for a problem known as percolation of words. Such extensions were already considered earlier, and we hope to combine known results with suitable adaptions of ideas and tools contained in our work on percolation of words.