  The gravity side of the gauge/gravity duality predicts the existence of small
black holes with negative specific heat. A free theory of strings has a
Hagedorn behavior, but it does not lead to negative specific heat. To
understand such states one needs to consider a theory of interacting strings.
In the dual gauge theory, the string interactions are related to non-planar
diagrams. In this paper the simplest gauge matrix model of two free matrices,
that has Hagedorn behavior is analyzed in detail. A simple double trace
deformation of the Hamiltonian, proportional to square of the free Hamiltonian
square with a negative sign that mimics a gravitational attraction is enough to
produce states with negative specific heat perturbatively and one can still
compute the equation of state relating the entropy and the energy. A more
general argument based on non-planar interactions that are random and that grow
faster in strength than the energy suggests that states with negative specific
heat appear generically.
