Parabolic Anderson Model: eigenvalue order statistics and path localization

Abstract: The Parabolic Anderson Model is a class of diffusion problems driven by the Anderson Hamiltonian \Delta+\xi, where \Delta is the Laplacian and \xi a random field. I will give an overview of recent results on this model in the case when the underlying spatial structure is the hypercubic lattice \mathbb Z^d and the \xi's are i.i.d. random variables. Then I will report on a recent progress in the understanding of the random walk associated, via the Feynman-Kac formula, to the diffusion equation \partial_t u=\Delta u+\xi u. Here Sznitman's technique of enlargement of obstacles generally shows that the walk up to time t localizes in one of t^{o(1)} localization centra within a ball of radius t^{1+o(1)}. In a specific class of potentials (double-exponential upper tail) I will show that there is typically only one such localization center and will characterize its distribution. The key step is the description of extreme order statistics of the eigenvalues of \Delta+\xi restricted to a large finite box. Joint work with W.K\"onig.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).