  We demonstrate that in the presence of Coulomb interactions, electrons in
graphene behave like a critical system, supporting power law correlations with
interaction-dependent exponents. An asymptotic analysis shows that the origin
of this behavior lies in particle-hole scattering, for which the Coulomb
interaction induces anomalously close approaches. With increasing interaction
strength the relevant power law changes from real to complex, leading to an
unusual instability characterized by a complex-valued susceptibility in the
thermodynamic limit. Measurable quantities, as well as the connection to
classical two dimensional systems, are discussed.
