  During the past decade, phase-transition phenomena in the random
3-satisfiability (3-SAT) problem has been intensively studied by statistical
physics methods. In this work, we study the random 3-SAT problem by the
mean-field first-step replica-symmetry-broken cavity theory at the limit of
temperature $T\to 0$. The reweighting parameter $y$ of the cavity theory is
allowed to approach infinity together with the inverse temperature $\beta$ with
fixed ratio $r=y / \beta$. Focusing on the the system's space of satisfiable
configurations, we carry out extensive population dynamics simulations using
the technique of importance sampling and we obtain the entropy density $s(r)$
and complexity $\Sigma(r)$ of zero-energy clusters at different $r$ values. We
demonstrate that the population dynamics may reach different fixed points with
different types of initial conditions. By knowing the trends of $s(r)$ and
$\Sigma(r)$ with $r$, we can judge whether a certain type of initial condition
is appropriate at a given $r$ value. This work complements and confirms the
results of several other very recent theoretical studies.
