  A number of recent numerical investigations concluded that the remnants of
rare structures formed at very high redshift, such as the very first stars and
bright redshift z~6 QSOs, are preferentially located at the center of the most
massive galaxy clusters at redshift z=0. In this paper we readdress this
question using a combination of cosmological simulations of structure formation
and extended Press-Schechter formalism and we show that the typical remnants of
Population III stars are instead more likely to be found in a group
environment, that is in dark matter halos of mass ~2x10^{13} h^{-1}M_sun.
Similarly, the descendants of the brightest z~6 QSOs are expected to be in
medium-sized clusters (mass of a few 10^{14} h^{-1}M_sun), rather than in the
most massive superclusters (M>10^{15} h^{-1}M_sun) found within the typical 1
Gpc^3 cosmic volume where a bright z~6 QSO lives. The origin of past claims
that the most massive clusters preferentially host these remnants is rooted in
the numerical method used to initialize their numerical simulations: Only a
small region of the cosmological volume of interest was simulated with
sufficient resolution to identify low-mass halos at early times, and this
region was chosen to host the most massive halo in the cosmological volume at
late times. The conclusion that the earliest structures formed in the entire
cosmological volume evolve into the most massive halo at late times was thus
arrived at by construction. We demonstrate that, to the contrary, the first
structures to form in a cosmological region evolve into relatively typical
objects at later times. We propose alternative numerical methods for simulating
the earliest structures in cosmological volumes.
