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An Improved Decoupling Inequality for Random Interlacements

  • Diego F. de Bernardini
  • Christophe GallescoEmail author
  • Serguei Popov
Article
  • 23 Downloads

Abstract

In this paper we obtain a decoupling feature of the random interlacements process \({{\mathcal {I}}}^u\subset {\mathbb {Z}}^d\), at level u, \(d\ge 3\). More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, \({\textsf {F}}\) and its translated \({\textsf {F}}+x\), can be coupled with high probability of success, when \(\Vert x\Vert \) is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two [0, 1]-valued functions depending on the configuration of the random interlacements on \({\textsf {F}}\) and \({\textsf {F}}+x\), respectively. This improves a previous bound obtained by Sznitman (Ann Math 2(171):2039–2087, 2010).

Keywords

Random interlacements Independent excursions Soft local times Decoupling 

Mathematics Subject Classification

60K35 60G50 82C41 

Notes

Acknowledgements

Diego F. de Bernardini was partially supported by São Paulo Research Foundation (FAPESP) (Grant 2014/14323-9). Christophe Gallesco was partially supported by FAPESP (Grant 2017/19876-4) and CNPq (Grant 312181/2017-5). Serguei Popov was partially supported by CNPq (Grant 300886/2008–0). The three authors were partially supported by FAPESP (Grant 2017/02022-2 and Grant 2017/10555-0).

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Statistics, Institute of Mathematics, Statistics and Scientific ComputationUniversity of Campinas - UNICAMPCampinasBrazil

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