Large deviations from the macroscopic equation for a particle systems with external random field and Kac type interaction

We consider a lattice gas in a periodic $d-$ dimensional lattice of width $\g^{-1}$, $\g>0$, interacting via a Kac's type interaction, with range $\frac 1\g $ and strength $\g^d$, and under the influence of a random potential given by independent, bounded, random variables with translational invariant distribution. The system evolves through a conservative dynamics, i.e. particles jump to nearest neighbor empty sites, with rates satisfying detailed balance with respect to the equilibrium measures. In [21] it has been shown that rescaling space as $\g^{-1}$ and time as $\g^{-2}$, in the limit $\g \downarrow 0$, for dimensions $d\ge 3$, the macroscopic density profile $\r$ satisfies, a.s. with respect to the random field, a nonlinear integral partial differential equation, having the diffusion matrix determined by the statistical properties of the external random field. Here we show an almost sure (with respect to the random field) large deviations principle for the empirical measures of such a process. The rate function, which depends on the statistical properties of the external random field, is lower semicontinuous and has compact level sets.