  We present evidence for a duality between Jackiw-Teitelboim gravity minimally
coupled to a free massive scalar field and a single-trace two-matrix model. One
matrix is the Hamiltonian $H$ of a holographic disorder-averaged quantum
mechanics, while the other matrix is the light operator $\cal O$ dual to the
bulk scalar field. The single-boundary observables of interest are thermal
correlation functions of $\cal O$. We study the matching of the genus zero one-
and two-boundary expectation values in the matrix model to the disk and
cylinder Euclidean path integrals. The non-Gaussian statistics of the matrix
elements of $\cal O$ correspond to a generalization of the ETH ansatz.
  We describe multiple ways to construct double-scaled matrix models that
reproduce the gravitational disk correlators. One method involves imposing an
operator equation obeyed by $H$ and $\cal O$ as a constraint on the two
matrices. Separately, we design a model that reproduces certain double-scaled
SYK correlators that may be scaled once more to obtain the disk correlators.
  We show that in any single-trace, two-matrix model, the genus zero
two-boundary expectation value, with up to one $\cal O$ insertion on each
boundary, can be computed directly from all of the genus zero one-boundary
correlators. Applied to the models of interest, we find that these cylinder
observables depend on the details of the double-scaling limit. To the extent we
have checked, it is possible to reproduce the gravitational double-trumpet,
which is UV divergent, from a systematic classification of matrix model `t
Hooft diagrams. The UV divergence indicates that the matrix integral saddle of
interest is perturbatively unstable. A non-perturbative treatment of the matrix
models discussed in this work is left for future investigations.
