  We study moderate deviations in the exponential corner growth model, both in
the bulk setting and the increment-stationary setting. The main results are
sharp right-tail bounds on the last-passage time and the exit point of the
increment-stationary process. The arguments utilize calculations with the
stationary version and a moment generating function identity due to E. Rains,
for which we give a short probabilistic proof. As applications of the deviation
bounds, we derive upper bounds on the speed of distributional convergence in
the Busemann function and competition interface limits.
