A two cities theorem for the parabolic Anderson model

The parabolic Anderson problem is the Cauchy problem for the heat equation ${\mathrm{\partial }}_{t}u(t,z)=\mathrm{\Delta }u(t,z)+\xi (z)u(t,z)$ on $(0,\infty)\times {$\mathbb{Z}$}^d$withrandompotential$(\xi(z):z\in{$\mathbb{Z}$}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\xi (z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,z)={\text{mathbh}1}_0(z)$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.