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Increasing propagation of chaos for mean field models. (English) Zbl 0928.60092
The authors consider mean-field measures $\mu^{(N)}$ with interacting potential $F$. The probability measure $\mu^{(N)}$ is the law of an interacting system of size $N$. The authors are interested in the asymptotic independence properties of these measures, as $N$ tends to infinity. This asymptotic behavior is called asymptotic chaoticity. Under mild assumptions, one proves that the empirical measures of the system converge in law to a deterministic measure $\mu^*$, and equivalently for any finite $k$, the convergence of the law of $k$-subsystem under $\mu^{(N)}$ to $(\mu^*)^{\otimes k}$. The authors estimate the relative entropy distance between $\mu^{(N)}$ and appropriate simpler exchangeable measures $\nu^N$ which are related to the law $\mu^*$. They prove that if the minimum of the function $H(\cdot\mid\mu)- F(\cdot)$ is unique, say $\mu^*$, then $\nu^N=(\mu^*)^{\otimes N}$, and propagation of chaos holds for blocks of size $o(N)$ if this minimum is nondegenerate. Certain degenerate situations are also studied. Next these results are applied to the Langevin dynamics of interacting particles leading to a McKean-Vlasov limit.

60K35Interacting random processes; statistical mechanics type models; percolation theory
82C22Interacting particle systems
60G99Stochastic processes
82C31Stochastic methods in time-dependent statistical mechanics
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