  A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT
has been undertaken. In combination with a monotonicity assumption we find that
the threshold for random 3-SAT satisfies $\alpha_3 \leq 4.262$. If the
assumption is correct, this means that the actual threshold value for $k=3$ is
lower than that given by the cavity method. In contrast the latter has recently
been shown to give the correct value for large $k$. Our result thus indicate
that there are distinct behaviors for $k$ above and below some critical $k_c$,
and the cavity method may provide a correct mean-field picture for the range
above $k_c$.
