Abstrakt
We study the non-negative solution to the Cauchy problem for the parabolic equation on with initial data. Here is the discrete Laplacian on and is an i.i.d. random field with doubly-exponential upper tails.
We prove that, for large t and with large probability, most of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes and are shown to converge in distribution under suitable scaling of space and time.
Aging results for , as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for in large sets recently proved by the first two authors.