Geometric characterization of intermittency in the parabolic Anderson model

We consider the parabolic Anderson problem ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+\xi (x)u$ on ${\mathbb{R}}_{+}\times {\mathbb{Z}}^d$ with localized initial condition $u(0,x)={\delta }_0(x)$ and random i.i.d. potential $\xi$. Under the assumption that the distribution of $\xi (0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to \mathrm{\infty }$, the overwhelming contribution to the total mass ${\sum }_{x}u(t,x)$ 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 $\xi$ in a box of side length $2t{\log }^{2}t$ for which the (local) principal Dirichlet eigenvalue of the random operator $\mathrm{\Delta }+\xi$ is close to the top of the spectrum in the box. We also prove that the shape of $\xi$ in these regions is nonrandom and that $u(t,\cdot )$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.