  Exact thermal studies of small (4-site, 5-site and 8-site)
  Hubbard clusters with local electron repulsion yield intriguing insight into
phase separation, charge-spin separation, pseudogaps, condensation, in
particular, pairing fluctuations away from half filling (near optimal doping).
These exact calculations, carried out in canonical (i.e. for fixed electron
number N) and grand canonical (i.e. fixed chemical potential $\mu$) ensembles,
monitoring variations in temperature T and magnetic field h, show rich phase
diagrams in a T-$\mu$ space consisting of pairing fluctuations and signatures
of condensation. These electron pairing instabilities are seen when the onsite
Coulomb interaction U is smaller than a critical value U$_c$(T) and they point
to a possible electron pairing mechanism. The specific heat, magnetization,
charge pairing and spin pairing provide strong support for the existence of
competing (paired and unpaired) phases near optimal doping in these clusters as
observed in recent experiments in doped La$_{2-x}$Sr$_x$CuO$_{4+y}$ high T$_c$
superconductors.
