Quantitative Estimates for Mean-Field Limits and Correlation Functions through a Duality Framework

TL;DR

This paper develops a duality framework to quantitatively analyze the mean-field limit and correlation functions in interacting particle systems, achieving optimal convergence rates under certain regularity conditions.

Contribution

It introduces a novel duality-based approach to derive sharp convergence and correlation estimates, including the optimal rate, for mean-field limits.

Findings

Achieves a fluctuation-scale rate of (N^{-1/2}) for square-integrable interactions.

Recovers the optimal mean-field rate (N^{-1}) using an iterative hierarchy of dual cumulants.

Provides refined bounds on correlations and deviations from chaos via dual and direct cumulants.

Abstract

We investigate the mean-field limit for interacting particle systems through a duality-based framework and obtain quantitative estimates on the convergence of marginals as well as on correlation functions. In particular, for merely square-integrable interaction forces, we derive the natural fluctuation-scale rate $\mathcal{O}(N^{-1/2})$. By introducing an iterative argument on the hierarchy of dual cumulants, we leverage this bound to recover the optimal mean-field rate $\mathcal{O}(N^{-1})$ and to obtain robust estimates on the dual cumulants, at the expense of corresponding regularity assumptions on the interaction kernel. Finally, using the relation between dual and direct correlations, we transfer these bounds to direct cumulants, yielding refined information on correlations and deviations from chaos.