  In this work, we investigate the presence of the weak Lefschetz property
(WLP) and Hilbert functions for various types of random standard graded
Artinian algebras. If an algebra has the WLP then its Hilbert function is
unimodal.
  Using probabilistic models for random monomial algebras, our results and
simulations suggest that in each considered regime the Hilbert functions of the
produced algebras are unimodal with high probability. The WLP appears to be
present with high probability most of the time. However, we propose that there
is one scenario where the generated algebras fail to have the WLP with high
probability.
