An Improved Decoupling Inequality for Random Interlacements

  • Diego F. de Bernardini
  • Christophe GallescoEmail author
  • Serguei Popov


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).


Random interlacements Independent excursions Soft local times Decoupling 

Mathematics Subject Classification

60K35 60G50 82C41 



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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