Proof of Triangle Condition: The Infrared Bound

  • Markus HeydenreichEmail author
  • Remco van der Hofstad
Part of the CRM Short Courses book series (CRMSC)


We state one of the key results in high-dimensional percolation, the so-called infrared bound that bounds the Fourier transform of the critical two-point function close to its singularity. This key result was first proved by Hara and Slade. We extend the discussion to a slightly modified percolation model known as spread-out percolation that signals the role of the upper-critical dimension more clearly than the usual nearest-neighbor model. We give an overview of the proof of the infrared bound that relies on the lace expansion. This proof stretches over the next three chapters. Finally, we discuss random walk triangles. The main idea is that if the random walk triangle is sufficiently small, then the infrared bound follows.

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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