Wasserstein-infinity stability and mean field limit of discrete interaction energy minimizers
Abstract
In this paper we give a quantitative stability result for the discrete interaction energy on the multi-dimensional torus, for the periodic Riesz potential. It states that if the number of particles $N$ is large and the discrete interaction energy is low, then the particle distribution is necessarily close to the uniform distribution (i.e., the continuous energy minimizer) in the Wasserstein-infinity distance. As a consequence, we obtain a quantitative mean field limit of interaction energy minimizers in the Wasserstein-infinity distance. The proof is based on the application of the author's previous joint work with J. Wang on the stability of continuous energy minimizer, together with a new mollification trick for the empirical measure in the case of singular interaction potentials.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.18395
- arXiv:
- arXiv:2407.18395
- Bibcode:
- 2024arXiv240718395S
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Classical Analysis and ODEs;
- 52C35;
- 74G65