We consider the parabolic Anderson problem ∂tu=Δu+ξ(x)u on ℝ+×ℤd 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) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑xu(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 of side length 2t log2t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.
"Geometric characterization of intermittency in the parabolic Anderson model." Ann. Probab. 35 (2) 439 - 499, March 2007. https://doi.org/10.1214/009117906000000764