# Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

## TL;DR

This paper derives the precise second-order asymptotic expansion for multimarginal optimal transport problems with Coulomb and Riesz costs, extending previous results and introducing a unified decomposition method applicable to a broad class of singular, long-range interactions.

## Contribution

It provides the first sharp second-order asymptotics for N-marginal optimal transport with Riesz costs, generalizing prior Coulomb-specific results and developing a robust Fefferman-Gregg decomposition for these kernels.

## Key findings

- Determined the second-order term in the N→∞ asymptotics for Riesz costs with 0<s<d.
- Proved a small oscillations property for the second-order energy.
- Extended the Fefferman-Gregg decomposition to a wider class of kernels.

## Abstract

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.