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Communications in Mathematical Physics

, Volume 362, Issue 2, pp 513–546 | Cite as

The Sign Clusters of the Massless Gaussian Free Field Percolate on \({\mathbb{Z}^{d}, d \geqslant 3}\) (and more)

  • Alexander Drewitz
  • Alexis Prévost
  • Pierre-Françcois Rodriguez
Article
  • 31 Downloads

Abstract

We investigate the percolation phase transition for level sets of the Gaussian free field on \({\mathbb{Z}^{d}}\), with \({d \geqslant 3}\), and prove that the corresponding critical parameter h*(d) is strictly positive for all \({d \geqslant 3}\), thus settling an open question from (Rodriguez and Sznitman in Commun Math Phys 320(2):571–601, 2013). In particular, this implies that the sign clusters of the Gaussian free field percolate on \({\mathbb{Z}^{d}}\), for all \({d \geqslant 3}\). Among other things, our construction of an infinite cluster above small, but positive level h involves random interlacements at level u > 0, a random subset of \({\mathbb{Z}^{d}}\) with desirable percolative properties, introduced in Sznitman (Ann Math (2) 171(3):2039–2087, 2010) in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field (Sznitman in Electron Commun Probab 17(9):9, 2012), and a recent coupling of these two objects (Lupu in Ann Probab 44(3):2117–2146, 2016), lifted to a continuous metric graph structure over \({\mathbb{Z}^{d}}\).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany
  2. 2.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

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