  The crossover between a metal and a Mott insulator leads to a localization of
fermions from delocalized Bloch states to localized states. We experimentally
study this crossover using fermionic atoms in an optical lattice by measuring
thermodynamic and local (on--site) density correlations. In the metallic phase
at incommensurable filling we observe the violation of the local
fluctuation--dissipation theorem indicating that the thermodynamics cannot be
explained by local observables. In contrast, in the Mott-insulator we observe
the convergence of local and thermodynamic fluctuations indicating the absence
of long--range density-density correlations.
