Abstract
We address the issue of the Central Limit Theorem for (both local and global) empirical measures of diffusions interacting on a possibly diluted Erdős-Rényi graph. Special attention is given to the influence of initial condition (not necessarily i.i.d.) on the nature of the limiting fluctuations. We prove in particular that the fluctuations remain the same as in the mean-field framework when the initial condition is chosen independently from the graph. We give an example of non-universal fluctuations for carefully chosen initial data that depends on the graph. A crucial tool for the proof is the use of extensions of Grothendieck inequality.
Acknowledgments
We would like to warmly thank the reviewers for their very thorough reading of our manuscript and for their valuable comments. C.P. thanks Guillaume Aubrun for very fruitful discussions and in particular for his help for the proof of Proposition 3.5. E.L. would like to thank warmly Jérôme Dedecker and Nathael Gozlan for useful discussions and acknowledges the support of ANR-19-CE40-002 (ChaMaNe). C.P. acknowledges the support of ANR-17-CE40-0030 (EFI). E.L. and C.P. both acknowledge the support of ANR–19–CE40–0023 (PERISTOCH).
Citation
Fabio Coppini. Eric Luçon. Christophe Poquet. "Central Limit Theorems for global and local empirical measures of diffusions on Erdős-Rényi graphs." Electron. J. Probab. 28 1 - 63, 2023. https://doi.org/10.1214/23-EJP1038
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