  Motivated by a problem arising from Density Functional Theory, we provide the
sharp next-order asymptotics for a class of multimarginal optimal transport
problems with cost given by singular, long-range pairwise interaction
potentials. More precisely, we consider an $N$-marginal optimal transport
problem with $N$ equal marginals supported on $\mathbb R^d$ and with cost of
the form $\sum_{i\neq j}|x_i-x_j|^{-s}$. In this setting we determine the
second-order term in the $N\to\infty$ asymptotic expansion of the minimum
energy, for the long-range interactions corresponding to all exponents $0<s<d$.
We also prove a small oscillations property for this second-order energy term.
Our results can be extended to a larger class of models than power-law-type
radial costs, such as non-rotationally-invariant costs. The key ingredient and
main novelty in our proofs is a robust extension and simplification of the
Fefferman-Gregg decomposition (Fefferman 1985, Gregg 1989), extended here to
our class of kernels, and which provides a unified method valid across our full
range of exponents. Our first result generalizes a recent work of Lewin, Lieb
and Seiringer (2017), who dealt with the second-order term for the Coulomb case
$s=1,d=3$, by different methods.
