Eigenvalue order statistics and mass concentration in the parabolic Anderson model

We study the non-negative solution $u=u(x,t)$ to the Cauchy problem for the parabolic equation $\partial_t u=\Delta u+\xi u$ on $\mathbb Z^d\times[0,\infty)$ with initial data $u(x,0)=1_0(x)$.

Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ 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 $U(t)=\sum_x u(x,t)$ 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 $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto frac1t \log U(t)$ 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. In the proof, we prove and use the characterization of eigenvalue order statistics for $\Delta+\xi$ in large boxes and the exponential localisation of the corresponding eigenvectors. (Joint work with Marek Biskup and Renato dos Santos).