  We establish Tracy-Widom asymptotics for the partition function of a random
polymer model with gamma-distributed weights recently introduced by
Sepp\"al\"ainen. We show that the partition function of this random polymer can
be represented within the framework of the geometric RSK correspondence and
consequently its law can be expressed in terms of Whittaker functions. This
leads to a representation of the law of the partition function which is
amenable to asymptotic analysis. In this model, the partition function plays a
role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of
random matrix theory.
