  Recent results on the annulus partition function in Liouville field theory
are applied to non-critical string theory, both below and above the critical
dimension. Liouville gravity coupled to $c\le 1$ matter has a dual formulation
as a matrix model. Two well-known matrix model results are reproduced precisely
using the worldsheet formulation: (1) the correlation function of two
macroscopic loops, and (2) the leading non-perturbative effects. The latter
identifies the eigenvalue instanton amplitudes of the matrix approach with disk
instantons of the worldsheet approach, thus demonstrating that the matrix model
is the effective dynamics of a D-brane realization of $d\le 1$ non-critical
string theory. In the context of string theory above the critical dimension,
i.e. $d\ge 25$, Liouville field theory realizes two-dimensional de Sitter
gravity on the worldsheet. In this case, appropriate D-brane boundary
conditions on the annulus realize the S-matrix for two-dimensional de Sitter
gravity.
