  When double neutron star or neutron star-black hole binaries merge, the final
remnant may comprise a central solar-mass black hole surrounded by a 0.01-0.1
solar masses torus. The subsequent evolution of this disc may be responsible
for short gamma-ray bursts (SGRBs). A comparable amount of mass is ejected into
eccentric orbits and will eventually fall back to the merger site after
approximately 0.01 seconds. In this Paper, we investigate analytically the fate
of the fallback matter, which may provide a luminous signal long after the disc
is exhausted. We find that matter in the eccentric tail returns at a
super-Eddington rate and is eventually (0.1 sec) unable to cool via neutrino
emission and accrete all the way to the black hole. Therefore, contrary to
previous claims, our analysis suggests that fallback matter is not an efficient
source of late time accretion power and is unlikely to cause the late flaring
activity observed in SGRB afterglows. The fallback matter rather forms a
radiation-driven wind or a bound atmosphere. In both cases, the emitting plasma
is very opaque and photons are released with a degraded energy in the X-ray
band. We therefore suggest that compact binary mergers could be followed by an
"X-ray renaissance", as late as several days to weeks after the merger. This
might be observed by the next generation of X-ray detectors.
