Convergence rate of entropy-regularized multi-marginal optimal transport costs

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter varepsilon tends to 0. We establish lower and upper bounds on the difference with the unregularized cost of the form Cvarepsilonlog(1/varepsilon)+O(varepsilon) for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semi-concave costs for a finer estimate, and lower bounds for mathscrC2 costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.