  We prove that under n^{1/3} scaling, the limiting distribution as n goes to
infinity of the free energy of Seppalainen's log-Gamma discrete directed
polymer is GUE Tracy-Widom. The main technical innovation we provide is a
general identity between a class of n-fold contour integrals and a class of
Fredholm determinants. Applying this identity to the integral formula proved in
[Corwin-O'Connell-Seppalainen-Zygouras] for the Laplace transform of the
log-Gamma polymer partition function, we arrive at a Fredholm determinant which
lends itself to asymptotic analysis (and thus yields the free energy limit
theorem). The Fredholm determinant was anticipated in [Borodin-Corwin] via the
formalism of Macdonald processes yet its rigorous proof was so far lacking
because of the nontriviality of certain decay estimates required by that
approach.
