  We take a step towards the non-perturbative description of a two-dimensional
dilaton-gravity theory which has a vanishing cosmological constant and contains
black holes. This is done in terms of a double-scaled Hermitian random matrix
model which non-perturbatively computes the partition function for the
asymptotic Bondi Hamiltonian. To arrive at this connection we first construct
the gauge-invariant asymptotic phase space of the theory and determine the
relevant asymptotic boundary conditions, compute the classical S-matrix and,
finally, shed light on the interpretation of the Euclidean path integral
defined in previous works. We then construct a matrix model that matches the
topological expansion of the latter, to all orders. This allows us to compute
the fine-grained Bondi spectrum and other late time observables and to
construct asymptotic Hilbert spaces. We further study aspects of the
semi-classical dynamics of the finite cut-off theory coupled to probe matter
and find evidence of maximally chaotic behavior in out-of-time-order
correlators. We conclude with a strategy for constructing the non-perturbative
S-matrix for our model coupled to probe matter and comment on the treatment of
black holes in celestial holography.
