  At zero temperature the lowest part of the spectrum of the QCD Dirac operator
is known to consist of delocalized modes that are described by random matrix
statistics. In the present paper we show that the nature of these eigenmodes
changes drastically when the system is driven through the finite temperature
cross-over. The lowest Dirac modes that are delocalized at low temperature
become localized on the scale of the inverse temperature. At the same time the
spectral statistics changes from random matrix to Poisson statistics. We
demonstrate this with lattice QCD simulations using 2+1 flavors of light
dynamical quarks with physical masses. Drawing an analogy with Anderson
transitions we also examine the mobility edge separating localized and
delocalized modes in the spectrum. We show that it scales in the continuum
limit and increases sharply with the temperature.
