Geometric Characterization of Intermittency in the Parabolic Anderson Model

We consider the parabolic Anderson problem ∂tu = ∆u + ξ(x)u on R+ × Z with localized initial condition u(0, x) = δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) lies in the vicinity of, or beyond, the double-exponential distribution, we prove the following geometric characterisation of intermittency: with probability one, as t → ∞, the overwhelming contribution to the total mass ∑ x u(t, x) comes from a slowly increasing number of islands which are located far from each other. These islands are local regions of those high exceedances of the field ξ in a box with radius t log t for which the (local) principal Dirichlet eigenvalue of the random operator ∆+ ξ is close to maximal. We also prove that the shape of ξ in these regions is non-random and that u(t, ·) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.