Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

Abstract

ABSTRACT. We study the non-negative solution u = u(x, t) to the Cauchy problem for the parabolic equationHere ∆ is the discrete Laplacian on Z d and ξ = (ξ(z)) z∈Z d 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 U(t) := ∑ x u(x, t) of the solution resides in a bounded neighborhood of a site Z t that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian ∆ + ξ and the distance to the origin. The processes t → Z t and t → 1 t log U(t) are shown to converge in distribution under suitable scaling of space and time. Aging results for Z t , as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for ∆ + ξ in large sets recently proved by the first two authors.

Delocalising the parabolic Anderson model through partial duplication of the potential

“…In [14] this was shown to also be the case for a model that replaced the Laplacian with the generator of a trapped random walk. By contrast, in very recent work [3] it has been shown that in the double-exponential case the PAM localises on a single connected island, rather than on a single site. This has confirmed the long standing conjecture that, in the i.i.d.…”

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

“…Similar results hold for a discrete relative of this operator, namely the Anderson model on ℓ 2 (Z d ), see e.g. [36,7,57,16,79,109]. The reason why these models are amenable to a very precise analysis is the applicability of Brownian motion, respectively random walk techniques and Feynman-Kac functionals.…”

“…Now it is clear why the study of the Anderson Hamiltonian in [16] gave also results on the (adjacency) percolation Laplacian. Upper bounds on the IDS N BA λ imply upper estimates for the IDS of the adjacency and Dirichlet site percolation model.…”

“…The next theorem concerns the Laplace operator on the full Cayley graph. It establishes a Lifshitz-type asymptotics in the sense that (16) lim…”

“…The paper [16], which studies the intermittency behaviour of the parabolic Anderson model, establishes also Lifshitz tails for the quantum site percolation model. Given the abovementioned relation between the Anderson and the quantum percolation Hamiltonian, it is not surprising that a detailed analysis of the former gives also results about the latter.…”