Submatrix deconfinement and small black holes in AdS

Large $N$ gauged multi-matrix quantum mechanical models usually have a first order Hagedorn transition, related to deconfinement. In this transition the change of the energy and entropy is of order $N^2$ at the critical temperature. This paper studies the microcanonical ensemble of the model at intermediate energies $1<<E<<N^2$ in the coexistence region for the first order phase transition. Evidence is provided for a partial deconfinement phase where submatrix degrees of freedom for a $U(M)$ subgroup of $U(N)$, with $M<<N$ have an excitation energy of order $M^2$ and are effectively phase separated from the other degrees of freedom. These results also provide a simple example of the Susskind-Horowitz-Polchinski correspondence principle where a transition from a long string to a black hole is smooth. Implications for the dual configurations of small black holes in $AdS$ are discussed.