  We prove that the two dimensional Hubbard model at finite temperature T and
half-filling is analytic in the coupling constant in a radius at least $c/(\log
T)^2$. We also study the self-energy through a new two-particle irreducible
expansion and prove that this model is not a Fermi liquid, but a Luttinger
liquid with logarithmic corrections. The techniques used are borrowed from
constructive field theory so the result is mathematically rigorous and
completely non-perturbative.
  Together with earlier results on the existence of two dimensional Fermi
liquids, this new result proves that the nature of interacting Fermi systems in
two dimensions depends on the shape of the Fermi surface.
