Probability and Stochastic Analysis Seminar

Decoupling inequalities for cylinders’ percolation

The cylinder’s percolation model arises from a Poissonian soup of infinite lines in $R^d$ and it is a stationary process under the isometries of the underlying space. Each such line is then thickened, becoming the axis of a cylinder of radius one. The associated percolation picture exhibits long range correlations and the rigidity of the underlying objects hampers direct attempts at proving decorrelation inequalities via sprinkling of the intensity parameter. We obtain such inequalities by exploiting the continuity of the process, taking the radii of the cylinders as a parameter and using it in a sprinkling argument. As an application, we prove that for small intensities of the cylinder’s process the simple random walk on the vacant set is transient. The talk will also go over similar decoupling inequalities for other models, their applications and open problems.
This talk is based on a joint work with Caio Alves.