  We compute exact second-order asymptotics for the cost of an optimal solution
to the entropic optimal transport problem in the continuous-to-discrete, or
semi-discrete, setting. In contrast to the discrete-discrete or
continuous-continuous case, we show that the first-order term in this expansion
vanishes but the second-order term does not, so that in the semi-discrete
setting the difference in cost between the unregularized and regularized
solution is quadratic in the inverse regularization parameter, with a leading
constant that depends explicitly on the value of the density at the points of
discontinuity of the optimal unregularized map between the measures. We develop
these results by proving new pointwise convergence rates of the solutions to
the dual problem, which may be of independent interest.
