Sharp phase transition in discrete and continuum percolation.

Speaker : Sanjoy Kumar Jhawar
Inria
Date: 17/01/2025
Time: 11:00 am - 12:00 pm
Location: Amphi 6

Abstract

This talk consists of simple example of percolation models: Bernoulli bond and site percolation on lattice and Poisson Boolean model on Euclidean space. In these models we say that percolation happens, if there exists a critical value of the parameter such that, the system has infinite component above the critical value of the parameter and all the components are finite if it is below the critical value. In this talk we shall explore the concept of sharp phase transition in the above percolation models using the seminal works by Hugo Duminil-Copin and co-authors. The term “sharp phase transition” is about how the probability of having a path from origin to a distance $n$, decays in $n$, in the sub-critical regime. It also captures how the percolation probability behaves in the supercritical regime. Later in this talk I will present some example of higher dimensional percolation models on the lattice on Z^d, namely the hypercube percolation and hole percolation model. The first model can be thought of as a generalization of the usual bond/site percolation model. Hiraoka and Mikami was the first to define the hole graph model. We will try to see how we can use the “OSSS method” developed by Hugo Duminil-Copin and co-authors, to prove sharp phase transition.