Full asymptotics for the parabolic Anderson model with Pareto potential in the heaviest-tailed regime

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case where $\{\xi (z):z\in {\mathbb{Z}}^d\}$ are independent and identically distributed Pareto random variables with parameter $\alpha$, and assume that the solution is initially localised at the origin. We establish the full asymptotic behaviour of the total mass of the solution as time tends to infinity in the heaviest-tailed regime $\alpha \in (d,2d)$. In particular, we find qualitatively different behaviour in dimension $d=1$ and in dimensions $d\ge 2$.