  We consider a quantum lattice system with infinite-dimensional on-site
Hilbert space, very similar to the Bose-Hubbard model. We investigate many-body
localization in this model, induced by thermal fluctuations rather than
disorder in the Hamiltonian. We provide evidence that the Green-Kubo
conductivity $\kappa(\beta)$, defined as the time-integrated current
autocorrelation function, decays faster than any polynomial in the inverse
temperature $\beta$ as $\beta \to 0$. More precisely, we define approximations
$\kappa_{\tau}(\beta)$ to $\kappa(\beta)$ by integrating the current-current
autocorrelation function up to a large but finite time $\tau$ and we rigorously
show that $\beta^{-n}\kappa_{\beta^{-m}}(\beta)$ vanishes as $\beta \to 0$, for
any $n,m \in \mathbb{N}$ such that $m-n$ is sufficiently large.
