Wasserstein convergence rates of increasingly concentrating probability measures

For $\mathrm{\ell }\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to [0,\mathrm{\infty })$ we consider the sequence of probability measures ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$, where ${\mu }_n$ is determined by a density that is proportional to $\exp (-n\mathrm{\ell })$. We allow for infinitely many global minimal points of $\mathrm{\ell }$, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the $p$-Wasserstein convergence of ${\left({\mu }_n\right)}_{n\in \mathbb{N}}$ to its limit measure. Imposing regularity conditions we obtain a speed of convergence of $n^{-1\mathrm{/}(2p)}$ and adding a further technical assumption, we can improve this to a $p$-independent rate of $1\mathrm{/}2$ for all orders $p\in \mathbb{N}$ of the Wasserstein distance.