Equality of critical parameters for percolation of Gaussian free field levelsets
Abstract
We consider levelsets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given realvalued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely $h_{**}(d)$, $h_{*}(d)$ and $\bar h(d)$, respectively describing a wellordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. $h_{**}(d)=h_{*}(d)= \bar h(d)$ for any $d \geq 3$. At the core of our proof lies a new interpolation scheme aimed at integrating out the longrange dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control socalled largefield effects. This approach opens the way to a complete understanding of the offcritical phases of strongly correlated percolation models.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.07735
 arXiv:
 arXiv:2002.07735
 Bibcode:
 2020arXiv200207735D
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60K35;
 82B43;
 60G15;
 60G60
 EPrint:
 54 pages, 5 figures