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 ${\partial }_{t}u=\mathrm{\Delta }u+\xi u$ on ${\mathbb{Z}}^d\times [0,\mathrm{\infty })$ with initial data $u(x,0)=1_0(x)$.

Here $\mathrm{\Delta }$ 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 $\mathrm{\Delta }+\xi$ 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 $\mathrm{\Delta }+\xi$ in large boxes and the exponential localisation of the corresponding eigenvectors. (Joint work with Marek Biskup and Renato dos Santos).