  We present an analytic expression of the nonperturbative free energy of a
double-well supersymmetric matrix model in its double scaling limit, which
corresponds to two-dimensional type IIA superstring theory on a nontrivial
Ramond-Ramond background. To this end we draw upon the wisdom of random matrix
theory developed by Tracy and Widom, which expresses the largest eigenvalue
distribution of unitary ensembles in terms of a Painleve II transcendent.
Regularity of the result at any value of the string coupling constant shows
that the third-order phase transition between a supersymmetry-preserving phase
and a supersymmetry-broken phase, previously found at the planar level, becomes
a smooth crossover in the double scaling limit. Accordingly, the supersymmetry
is always broken spontaneously as its order parameter stays nonzero for the
whole region of the coupling constant. Coincidence of the result with the
unitary one-matrix model suggests that one-dimensional type 0 string theories
partially correspond to the type IIA superstring theory. Our formulation
naturally allows for introduction of an instanton chemical potential, and
reveals the presence of a novel phase transition, possibly interpreted as
condensation of instantons.
