  We study a (1+1)-dimensional directed polymer in a random environment on the
integer lattice with log-gamma distributed weights. Among directed polymers,
this model is special in the same way as the last-passage percolation model
with exponential or geometric weights is special among growth models, namely,
both permit explicit calculations. With appropriate boundary conditions, the
polymer with log-gamma weights satisfies an analogue of Burke's theorem for
queues. Building on this, we prove the conjectured values for the fluctuation
exponents of the free energy and the polymer path, in the case where the
boundary conditions are present and both endpoints of the polymer path are
fixed. For the polymer without boundary conditions and with either fixed or
free endpoint, we get the expected upper bounds on the exponents.
