The sign clusters of the massless Gaussian free field percolate on ℤ^d, d \geqslant 3 (and more)

2017/08/31 by Alexander Drewitz, Alexis Prévost, Pierre-François Rodriguez
Mathematics · Physics and Astronomy · #math-ph #math.MP #math.PR #msc:60G15 #msc:60G60 #msc:60K35 #msc:82B43

paper · pdf · doi:10.1007/s00220-018-3209-6

published as Commun. Math. Phys. 362, 513-546 (2018) · 29 pages, final version, to appear in Comm. Math. Phys

arxiv created 2018/05/25 · arxiv updated 2018/08/29

Abstract

We investigate the percolation phase transition for level sets of the Gaussian free field on ℤ^d, with d\geqslant 3, and prove that the corresponding critical parameter h_*(d) is strictly positive for all d\geqslant3, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on 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 ℤ^d with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over ℤ^d.

Citations

• Phase transition and level-set percolation for the Gaussian free field
• Vacant Set of Random Interlacements and Percolation
• An isomorphism theorem for random interlacements