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Decoupling Inequalities

  • Alexander Drewitz
  • Balázs Ráth
  • Artëm Sapozhnikov
Chapter
  • 678 Downloads
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter we define subclasses of increasing and decreasing events for which the decorrelation inequalities (7.4.2) and (7.4.3) hold with rapidly decaying error term; see Sect. 8.1 and Theorem 8.3. The definition of these events involves a treelike hierarchical construction on multiple scales L n . In Sect. 8.2 we state the decoupling inequalities for a large number of local events; see Theorem 8.5. We prove Theorem 8.5 in Sect. 8.3 by iteratively applying Theorem 8.3. Finally, in Sect. 8.4, using Theorem 8.5 we prove that if the density of certain patterns in \({\mathcal{I}}^{u}\) is small, then it is very unlikely that such patterns will be observed in \({\mathcal{I}}^{u(1\pm \delta )}\) along a long path; see Theorem 8.7. This last result will be crucially used in proving that u ∈ (0, ) in Chap. 9.

Keywords

Random Interlacements Gaussian Free Field Dyadic Tree Monotonic Effect General Induction Step 
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Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Alexander Drewitz
    • 1
  • Balázs Ráth
    • 2
  • Artëm Sapozhnikov
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew York CityUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Max-Planck Institute of Mathematics in the SciencesLeipzigGermany

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