  We study the convergence of entropically regularized optimal transport to
optimal transport. The main result is concerned with the convergence of the
associated optimizers and takes the form of a large deviations principle
quantifying the local exponential convergence rate as the regularization
parameter vanishes. The exact rate function is determined in a general setting
and linked to the Kantorovich potential of optimal transport. Our arguments are
based on the geometry of the optimizers and inspired by the use of $c$-cyclical
monotonicity in classical transport theory. The results can also be phrased in
terms of Schr\"odinger bridges.
