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Journal of Statistical Physics

, Volume 176, Issue 5, pp 1057–1087 | Cite as

Interacting Diffusions on Random Graphs with Diverging Average Degrees: Hydrodynamics and Large Deviations

  • Roberto I. OliveiraEmail author
  • Guilherme H. Reis
Article

Abstract

We consider systems of mean-field interacting diffusions, where the pairwise interaction structure is described by a sparse (and potentially inhomogeneous) random graph. Examples include the stochastic Kuramoto model with pairwise interactions given by an Erdős–Rényi graph. Our problem is to compare the bulk behavior of such systems with that of corresponding systems with dense nonrandom interactions. For a broad class of interaction functions, we find the optimal sparsity condition that implies that the two systems have the same hydrodynamic limit, which is given by a McKean–Vlasov diffusion. Moreover, we also prove matching behavior of the two systems at the level of large deviations. Our results extend classical results of dai Pra and den Hollander and provide the first examples of LDPs for systems with sparse random interactions.

Keywords

McKean–Vlasov equations Mean-field models Random graphs Large deviations 

Notes

Acknowledgements

R. I. Oliveira: Supported by a Bolsa de Produtividade em Pesquisa from CNPq, Brazil. His work in this article is part of the activities of FAPESP Center for Neuromathematics (grant # 2013/07699-0, FAPESP - S. Paulo Research Foundation). G. H. Reis: Supported by a Ph.D. scholarship from CNPq, Brazil (grant # 140768/2015-7.)

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil

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