Decoupling Inequalities

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.