  A finite size scaling analysis of the spectral function and of the optical
conductivity of a single hole moving in an antiferromagnetic background is
performed. It is shown that both the low energy quasiparticle peak and the
broad higher energy structure are robust with increasing cluster size from
$4\times 4$ to $\sqrt{26}\times\sqrt{26}$ sites. In the abscence of spin
fluctuations, for most static or dynamical quantities saturation occurs when
the size exceeds a characteristic size $N_c(J_z)$. Typically, 16 and 26 site
clusters give reliable results for $J_z>0.75$ and $J_z>0.3$ respectively. The
hole optical mass is shown to be very large ($>20$) in agreement with the small
bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a
small gap $\propto J_z$ separates the low energy delta-function from the rest
of the spectrum in the dynamical correlation functions. On the other hand, with
$J_\perp$ this gap seems to disappear with increasing system size as one would
expect since the spin waves are gapless in the thermodynamic limit. The large
momentum dependence of the quasiparticle weight in the isotropic case is
inconsistent with a string picture but agrees well with the self-consistent
Born approximation. An accurate estimation of the higher energy part of the
spectral functions of the t--J model can be made for momenta close to $(0,0)$
or $(\pi,\pi)$;
