Fluctuations of the quadratic matching cost on the flat torus: dimensions five and higher

We prove a central limit theorem for the quadratic matching cost between the uniform measure and its empirical counterpart on the flat torus in dimensions $d\ge 5$. Let $\mu$ be Haar probability measure on ${\mathbb{T}}^d$ and let ${\mu }_n$ be the empirical measure of $n$ independent samples from $\mu$. Then $$\begin{array}{rl}{\displaystyle \frac{W_2^2({\mu }_n,\mu )-\mathbb{E}{W}_2^2({\mu }_n,\mu )}{\sqrt{\mathrm{Var}(W_2^2({\mu }_n,\mu ))}}}& {\displaystyle ⟹N(0,1),}\\ {\displaystyle c_d{n}^{-1-4\mathrm{/}d}\le \mathrm{Var}(W_2^2({\mu }_n,\mu ))}& {\displaystyle \le C_d{n}^{-1-4\mathrm{/}d},}\end{array}$$ for positive constants $c_d,C_d$ depending only on $d$ and all sufficiently large $n$. The proof introduces a random-walk approach to matching fluctuations. We construct a random walk that jumps between neighboring cells of the optimal partition and use it to describe how the matching cost responds to changes in the sample. Estimates for this walk control the effect of replacing one sample point and the interaction between two replacements. These bounds yield the variance estimates and allow us to apply a martingale central limit theorem.