  We consider the generalization of a matrix integral with arbitrary spectral
curve $\rho_0(E)$ to a 0+1D theory of matrix quantum mechanics (MQM). Using
recent techniques for 1D quantum systems at large-$N$, we formulate a
hydrodynamical effective theory for the eigenvalues. The result is a simple 2D
free boson BCFT on a curved background, describing the quantum fluctuations of
the eigenvalues around $\rho_0(E)$, which is now the large-$N$ limit of the
quantum expectation value of the eigenvalue density operator $\hat{\rho}(E)$.
The average over the ensemble of random matrices becomes a quantum expectation
value. Equal-time density correlations reproduce the results (including
non-perturbative corrections) of random matrix theory. This suggests an
interpretation of JT gravity as dual to a $\textit{one-time-point}$ reduction
of MQM. As an application, we compute the R\'enyi entropy associated to a
bipartition of the eigenvalues. We match a previous result by Hartnoll and
Mazenc for the $c=1$ matrix model dual to two-dimensional string theory and
extend it to arbitrary $\rho_0(E)$. The hydrodynamical theory provides a clear
picture of the emergence of spacetime in two dimensional string theory. The
entropy is naturally finite and displays a large amount of short range
entanglement, proportional to the microcanonical entropy. We also compute the
reduced density matrix for a subset of $n<N$ eigenvalues.
