Geometric characterization of intermittency in the parabolic Anderson model

We consider the parabolic Anderson problem @tu = u + (x)u on R+ Z d with localized initial condition u(0;x) = 0(x) and random i.i.d. potential . Under the assumption that the distribution of (0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterisation of intermittency: with probability one, as t ! 1, the overwhelming contribution to the total mass P 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 eld in a box of side length 2t log 2 t for which the (local) principal Dirichlet eigenvalue of the random operator + is close to the top of the spectrum in the box. We also prove that the shape of 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.