Cotar, C; Petrache, M; (2019) Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs. Advances in Mathematics , 344 pp. 137-233. 10.1016/j.aim.2018.12.008.

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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 Rd and with cost of the form ∑i≠j|xi−xj|−s. In this setting we determine the second-order term in the N→∞ 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 [20], [26], 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 [36], who dealt with the second-order term for the Coulomb case s=1,d=3.

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