Critical Exponents and the Triangle Condition

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


We define an important condition, the so-called triangle condition introduced by Aizenman and Newman, that implies mean-field behavior in percolation in the sense that the percolation critical exponents \(\beta , \gamma \) and \(\delta \) take on their mean-field values. We illustrate the use of the triangle condition in its simplest setting by proving that it implies that \(\gamma =1\), and prove one-sided mean-field bounds on the critical exponents \(\delta \) and \(\beta \).

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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