Ideas Seminar: Promit Ghosal, Brandeis

Title: Fractal Geometry of the Parabolic Anderson Model in higher dimension.

Abstract: Parabolic Anderson model (PAM) is one of the prototypical frameworks for modelling conduction of electrons in crystals filled with defects. Intermittency of the peaks of the PAM is one of the widely studied topics in the last few decades and it holds close ties with the phenomenon of Anderson localization.  We show that the peaks of the PAM in dimension 2 and 3 are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) of those peaks. As a byproduct, we obtain the exact spatial asymptotics of the solution of the PAM. We also study the spatio-temporal peaks of the PAM and show their macroscopic multifractality. Some of the major tools used in our proof techniques include paracontrolled calculus and tail probabilities of the largest point in the spectrum of the Anderson Hamiltonian. This talk is based on a joint work with Jaeyun Yi.