Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four

We establish sharp variance asymptotics and limiting distributions for the quadratic optimal matching cost between the uniform measure and an empirical measure counterpart on the flat torus ${\mathbb{T}}^d$ in dimensions $d=2,3,4$. More precisely, let $\mu$ be Haar probability measure and ${\mu }_n$ the empirical measure of $n$ independent samples drawn from $\mu$. Then, for $d=2,3$, $$\begin{array}{rl}{\displaystyle n(W_2^2({\mu }_n,\mu )-\mathbb{E}[W_2^2({\mu }_n,\mu )])}& {\displaystyle \stackrel{\mathrm{d}}{\to }{L}_d,}\\ {\displaystyle n^2\mathrm{Var}(W_2^2({\mu }_n,\mu ))}& {\displaystyle ⟶\frac{1}{8{\pi }^4}\sum _{k\in {\mathbb{Z}}^d\setminus \{0\}}\frac{1}{\mathrm{\mid }k{\mathrm{\mid }}^4},}\end{array}$$ where $L_d$ is an explicit non-Gaussian weighted sum of independent centered exponential random variables. In dimension four, $$\frac{n}{\sqrt{\log n}}(W_2^2({\mu }_n,\mu )-\mathbb{E}[W_2^2({\mu }_n,\mu )])\stackrel{\mathrm{d}}{\to }N(0,\frac{1}{16{\pi }^2})\mathrm{.}$$ The main idea of the proof is to rely on a Hoeffding decomposition of the optimal transport cost, which turns out to be asymptotically equivalent to U-statistics of order 2.