# Probability and Stochastic Analysis Seminar

## 01/02/2023, 16:00 — 17:00 — OnlineAugusto Teixeira, Instituto de Matemática Pura e Aplicada

### Phase transition for percolation with axes-aligned defects

In this talk we will review a model that was first introduced by Jonasson, Mossel and Peres. Starting with the usual square lattice on $Z^2$, entire rows (respectively columns) of edges extending along the horizontal (respectively vertical) direction are removed independently at random. On the remaining thinned lattice, Bernoulli bond percolation is performed, giving rise to a percolation model with infinite range dependencies under the annealed law. In 2005, Hoffman solved the main conjecture around this model: proving that this percolation process indeed undergoes a nontrivial phase transition. In this talk, besides reviewing this surprisingly challenging problem, we will present a novel proof, which replaces the dynamic renormalization presented previously by a static version. This makes the proof easier to follow and to extend to other models. We finally present some remarks on the sharpness of Hoffman’s result as well as a list of interesting open problems that we believe can provide a renewed interest in this family of questions. This talk is based on a joint work with M. Hilário, M. Sá and R. Sanchis.

Except for a few of the oldest sessions these are from the Seminário de Probabilidade e Mecânica Estatística at IMPA which is co-sponsored by other institutions, in particular Instituto Superior Técnico.