We propose a simple instanton-based random matrix model of hot QCD
that in the quenched case precisely reproduces the distribution of the
lowest lattice overlap Dirac eigenvalues. Even after including dynamical
quarks the model can be easily simulated in volumes and for quark masses
that will be out of reach for direct lattice simulations in the foreseeable
future. Our simulations show that quantities connected to the $U(1)_A$ and
$SU(N_f)_A$ chiral symmetry are dominated by eigenvalues in a peak of the
spectral density that becomes singular at zero in the thermodynamic
limit. This spectral peak turns out to be produced by an ideal instanton
gas. By generalizing Banks-Casher type integrals for the singular spectral
density, definite predictions can be given for physical quantities that are
essential to test chiral symmetry breaking, but presently impossible to
compute reliably with direct lattice simulations.