  The parabolic Anderson problem is the Cauchy problem for the heat equation
$\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times
{\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. 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 $u(0,{z})={\mathbh1}_0({z})$, 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.
