We study the solutions $u=u(x,t)$ to the Cauchy problem on $mathbb Z^dtimes(0,infty)$ for the parabolic equation $partial_t u=Delta u+xi u$ with initial data $u(x,0)=1_{{0}}(x)$. Here $Delta$ is the discrete Laplacian on $mathbb Z^d$ and $xi=(xi(z))_{zinmathbb Z^d}$ is an i.i.d. random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority 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 $tmapsto Z_t$ and $t mapsto tfrac1t log U(t)$ 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. The proof uses the characterization of eigenvalue order statistics for $Delta+xi$ in large sets recently proved by the first two authors.
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