Percolation models were originally introduced to describe the propagation of a fluid in a random medium. In dimension two, the percolation properties of a model are encoded by so-called crossing probabilities (probabilities that certain rectangles are crossed from left to right). In the eighties, Russo, Seymour and Welsh obtained general bounds on crossing probabilities for Bernoulli percolation (the most studied percolation model, where edges of a lattice are independently erased with some given probability $1-p$). These inequalities rapidly became central tools to analyze the critical behavior of the model.
In this talk I will present a new result which extends the Russo-Seymour-Welsh theory to general percolation measures satisfying two properties: symmetry and positive correlation. This is a joint work with Laurin Köhler-Schindler.
- Combinatorial Theory Seminar