  Within the path integral formalism, we compute the disk partition functions
of two dimensional Liouville and JT quantum gravity theories coupled to a
matter CFT of central charge $c$, with cosmological constant $\Lambda$, in the
limit $c\rightarrow -\infty$, $|\Lambda|\rightarrow\infty$, for fixed
$\Lambda/c$ and fixed and finite disk boundary length $\ell$, to leading and
first subleading order in the $1/|c|$ expansion. In the case of Liouville
theory, we find perfect agreement with the asymptotic expansion of the known
exact FZZT partition function. In the case of JT gravity, we obtain the first
explicit results for the partition functions at finite cut-off, in the three
versions (negative, zero and positive curvature) of the model. Our findings are
in agreement with predictions from the recent proposal for a microscopic
definition of JT gravity, including the $c\rightarrow -\infty$ expansion of the
Hausdorff dimension of the boundary. In the negative curvature case, we also
provide evidence for the emergence of an effective Schwarzian description at
length scales much greater than the curvature length scale.
