Equality of critical parameters for percolation of Gaussian free field level-sets

Abstract

We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued 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, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.

Figures (5)

• Figure 1: An illustration of the event $\mathcal{G}_n$: depicted is a pair of admissible sets $(S_1,S_2)$ and the good bridge (in light gray) connecting them. Later in Section \ref{['sec:decompose_GFF']}, the underlying good events will guarantee that the sets $S_1$ and $S_2$ can be linked by a certain path (in red) inside a good bridge. Albeit not required by the definition, our construction of a good bridge on certain good events actually yields a "croissant-type" shape. More precisely, one can define two sequences of boxes, starting with the $0$-boxes $B_1$ and $B_2$ intersecting $S_1$ and $S_2$ (see \ref{['B2']}), respectively, and corresponding to the two arches in the proof of Lemma \ref{['L:bridge1']}, which comprise all but the largest boxes involved in the bridge construction and whose side lengths are non-decreasing. The largest boxes in the bridge have side length $L_{n-1}$ and comprise the deck.
• Figure 2: Some of the clusters in $\mathcal{C}$. On the event $A$, these clusters are $u(N)$-dense in the annulus $V_{2j}\setminus V_{2j+2}$. Each of the boxes $\tilde{\Lambda}_{\ell}$ (grey) is intersected by both the clusters in the support of $\widetilde{\mathcal{U}}_1$ (black) and $\widetilde{\mathcal{U}}_2$ (red), see \ref{['Upartition']}; the picture corresponds to Case 2, i.e. $|\mathcal{U}_{j+\frac{1}{2},j+1}(\omega_j)| \neq 0$ in the arguments following \ref{['Uconnect']}--\ref{['Upartition']}. When $G$ occurs, each box $\Lambda_{\ell}$ provides the opportunity to link two admissible pieces of $\widetilde{C}_1$ and $\widetilde{C}_2$ using a good bridge at a cost given by Lemma \ref{['lem:sprinkling']}.
• Figure 3: The two interpolation curves used in \ref{['eq:comparison']}. The red curve demarcates the boundary of the region in which the family of differential inequalities \ref{['difinal']} hold.
• Figure 4: Decoupling in the proof of Lemma \ref{['lem:piv_decoupling']}. By forcing the event $E_x$ (in red), which is independent of $\mathcal{Z}(\Lambda_x)$ and not too costly since $h> \tilde{h}$, $F_x$ and $f(\psi_x)$ decouple.
• Figure 5: Finding a good scale and reconstructing. On the event $\mathrm{CoarsePiv}_x(8\kappa L_N)\cap\mathcal{G}_{N, x}$, one constructs a path (red) in $\{ \chi \geq h\}$ connecting the boundaries (dotted) of the clusters of $B_r$ and $B_R$. The point $z$ is flipped to open and $y$ becomes a pivotal point (Lemma \ref{['lem:good_bridge']}). The occurrence of $\mathcal{G}_{n,x}^c$, decoupled by a dual surface, balances the reconstruction cost from the bridge (Lemma \ref{['lem:piv_find_goodscale']}).

Theorems & Definitions (57)

• Theorem \oldthetheorem
• Corollary \oldthetheorem: Decay of the truncated two-point function except at criticality
• Proposition \oldthetheorem
• Proposition \oldthetheorem
• Proposition \oldthetheorem
• Lemma \oldthetheorem: Bridging lemma
• Definition \oldthetheorem: Bridge
• Definition \oldthetheorem: Good bridge
• Theorem \oldthetheorem
• Remark \oldthetheorem