  We present a gravitational theory that interpolates between JT gravity, and a
gravity theory with a fixed boundary Hamiltonian. For this, we consider a
matrix integral with the insertion of a Gaussian with variance $\sigma^2$,
centered around a matrix $\textsf{H}_0$. Tightening the Gaussian renders the
matrix integral less random, and ultimately it collapses the ensemble to one
Hamiltonian $\textsf{H}_0$. This model provides a concrete setup to study
factorization, and what the gravity dual of a single member of the ensemble is.
We find that as $\sigma^2$ is decreased, the JT gravity dilaton potential gets
modified, and ultimately the gravity theory goes through a series of phase
transitions, corresponding to a proliferation of extra macroscopic holes in the
spacetime. Furthermore, we observe that in the Efetov model approach to random
matrices, the non-averaged factorizing theory is described by one simple saddle
point.
