  We construct a random Schrodinger operator on a subset of the hexagonal
lattice and study its smallest positive eigenvalues. Using an asymptotic
mapping, we relate them to the partition function of the directed polymer model
on the square lattice. For a specific choice of the edge weight distribution,
we obtain a model known as the log-Gamma polymer, which is integrable. Recent
results about the fluctuations of free energy for the log-Gamma polymer allow
us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the
random Schrodinger operator. We also relate the distribution of its k smallest
positive eigenvalues to the nonintersecting partition functions of order k.
