  Spinless fermions with repulsion are treated non-perturbatively by
classifying the diagrams of the generating functional $\Phi$ in powers of the
inverse lattice dimension $1/d$. The equations derived from the first two
orders are evaluated on the one- and on the two-particle level. The order
parameter of the AB-charge density wave (AB-CDW) occurring at larger
interaction is calculated in $d=3$. The Bethe-Salpeter equation is evaluated
for the conductivity $\sigma(\om)$ which is found to have two peaks within the
energy gap $2\Delta$ in the AB-CDW: a remnant of the Drude peak and an
excitonic resonance. Unexpectedly, $\sigma_{\rm\scriptscriptstyle DC}$ does not
vanish for $T\to 0$
