  With the aid of both a semi-analytical and a numerically exact method we
investigate the charge dynamics in the vicinity of half-filling in the one- and
two-dimensional $t$-$J$ model derived from a Fermi-Hubbard model in the limit
of large interaction $U$ and hence small exchange coupling $J$. The spin
degrees of freedom are taken to be disordered. So we consider the limit $0 < J
\ll T \ll W$ where $W$ is the band width. We focus on evaluating the spectral
density of a single hole excitation and the charge gap which separates the
upper and the lower Hubbard band. One of the key findings is the evidence for
the absence of sharp edges of the Hubbard band, instead Gaussian tails appear.
