  We perform numerical simulations of the dynamical equations for free water
surface in finite basin in presence of gravity. Wave Turbulence (WT) is a
theory derived for describing statistics of weakly nonlinear waves in the
infinite basin limit. Its formal applicability condition on the minimal size of
the computational basin is impossible to satisfy in present numerical
simulations, and the number of wave resonances is significantly depleted due to
the wavenumber discreteness. The goal of this paper will be to examine which WT
predictions survive in such discrete systems with depleted resonances and which
properties arise specifically due to the discreteness effects. As in
\cite{DKZ,onorato,naoto}, our results for the wave spectrum agree with the
Zakharov-Filonenko spectrum predicted within WT. We also go beyond finding the
spectra and compute probability density function (PDF) of the wave amplitudes
and observe an anomalously large, with respect to Gaussian, probability of
strong waves which is consistent with recent theory \cite{clnp,cln}. Using a
simple model for quasi-resonances we predict an effect arising purely due to
discreteness: existence of a threshold wave intensity above which turbulent
cascade develops and proceeds to arbitrarily small scales. Numerically, we
observe that the energy cascade is very ``bursty'' in time and is somewhat
similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade
arrest due to discreteness leads to accumulation of energy near the forcing
scale which, in turn, leads to widening of the nonlinear resonance and,
therefore, triggering of the cascade draining the turbulence levels and
returning the system to the beginning of the cycle. ~
