  At low temperature in the epsilon regime of QCD the low-end of the Dirac
spectrum is described by random matrix theory. In contrast, there has been no
similarly well established staistical description in the high temperature,
chirally symmetric phase. Using lattice simulations we show that at high
temperature a band of extremely localized eigenmodes appear at the low-end of
the Dirac spectrum. The corresponding eigenvalues are statistically independent
and obey a generalized Poisson distribution. Higher up in the spectrum the
Poisson distribution rapidly crosses over into the bulk distribution predicted
by the random matrix ensemble with the corresponding symmetry. Our results are
based on quenched lattice simulations with the overlap and the staggered Dirac
operator done well above the critical temperature at several volumes and values
of $N_t$. We also discuss the crucial role played by the fermionic boundary
condition and the Polyakov-loop in this phenomenon.
