  We model the evolution of galaxy clustering through cosmic time to
investigate the nature of the power-law shape of xi(r), the galaxy two-point
correlation function. Departures from a power law are mainly governed by galaxy
pair counts on small scales, subject to non-linear dynamics. We assume that
galaxies reside within dark matter halos and subhalos and use a semi-analytic
substructure evolution model to study subhalo populations within host halos. We
find that tidal mass loss and, to a lesser degree, dynamical friction deplete
the number of subhalos within larger host halos over time by ~90%, just the
right amount for achieving a power-law xi(r) at z = 0. We find that xi(r)
breaks from a power law at high masses, implying that only galaxies of
luminosities <= Lstar should exhibit power-law clustering. We also demonstrate
that xi(r) evolves from being far from a power law at high redshift, toward a
near power law at z = 0 and deviates in the future. This is mainly caused by
the evolving competition between the accretion and destruction rates of
subhalos, which happen to strike just the right balance at z~0. We show that
key ingredients determining the shape of xi(r) are the fraction of galaxies
that are satellites, the relative difference in mass between the halos of
isolated galaxies and halos that contain a single satellite on average, and the
rareness of halos that host galaxies. These pieces are intertwined and we find
no simple, universal rule for which a power-law xi(r) will occur. However, we
do show that the physics responsible for setting the galaxy content of halos do
not care about the conditions needed to achieve a power law xi(r) and these
conditions are met only in a narrow mass and redshift range. We conclude that
the power-law nature of xi(r) for Lstar and fainter galaxy samples at low
redshift is a cosmic coincidence. (Abridged)
