  The evolution of halos with masses around $M_\textrm{h} \approx 10^{11}\;
\textrm{M}_\odot$ and $M_\textrm{h} \approx 10^{12}\; \textrm{M}_\odot$ at
redshifts $z>9$ is examined using constrained N-body simulations. The specific
mass accretion rates, $\dot{M}_\textrm{h} / M_\textrm{h}$, exhibit minimal mass
dependence and agree with existing literature. Approximately one-third of
simulations reveal an increase in $\dot{M}_\textrm{h}$ around $z\approx 13$,
possibly implying a dual-age stellar population. Comparing simulated halos with
observed galaxies having spectroscopic redshifts, we find that for galaxies at
$z\gtrsim9$, the ratio between observed star formation rate (SFR) and
$\dot{M}_\textrm{h}$ is approximately $2\%$. This ratio remains consistent for
the stellar-to-halo mass ratio (SHMR) but only for $z\gtrsim10$. At $z\simeq
9$, the SHMR is notably lower by a factor of a few. At $z\gtrsim10$, there is
an agreement between specific star formation rates (sSFRs) and
$\dot{M}_\textrm{h} / M_\textrm{h}$ of halos. However, at $z\simeq 9$, observed
sSFRs exceed simulated values by a factor of two. To explain the relatively
high star formation efficiencies in high-$z$ halos with $M_\textrm{h} \approx
10^{11} M_{\odot}$, a simplified model is proposed, assuming the applicability
of the local Kennicutt-Schmidt law. The enhanced efficiency relative to low-$z$
is mainly driven by the reduced effectiveness of stellar feedback due to deeper
gravitational potential for halos of a fixed mass.
