Cillian Doherty: The square summability of the CLE complementary component diameters

Abstract:  Conformal loop ensembles (CLE) are a one-parameter family of random collections of loops in the plane. They are conjectured (and in some cases proven) to arise as the scaling limit of a wide variety of discrete models in probability and statistical physics. \(\mathrm{CLE}_\kappa\) can be defined when \(\kappa\) is in \((8/3,8)\) and determines a random Sierpinski carpet called the CLE carpet (or gasket). We prove that the sum of the squares of the diameters of the connected components of the complement of the CLE carpet is almost surely finite for all \(\kappa\) in \((8/3, 8) \setminus {4}\). This is a prerequisite for the application of a result of Ntalampekos on the uniformization of Sierpiński carpets. This talk is based on joint work with Jason Miller.