Geometry of Gaussian free field sign clusters and random interlacements

TL;DR

This work establishes a robust phase-structure for Gaussian free field sign clusters on a broad class of transient amenable graphs by proving a strong supercritical regime in which exactly two infinite sign clusters exist (one per sign) and finite islands are exponentially rare for heights $h<\overline{h}$. It builds a deep link between GFF level-set geometry and the vacant set of random interlacements through a cable-system isomorphism, showing that the interlacement vacant set percolates at small intensity $u$ on these graphs (resolving an open problem). The authors develop a comprehensive multi-scale renormalization framework enabled by precise decoupling inequalities for both GFF and RI, and leverage a sign-flipping coupling to translate interlacement connectivity into GFF sign-cluster percolation, extending Euclidean lattice results to fractal and Cayley-graph geometries. The results apply to classic lattices like $\mathbb{Z}^d$ with $d\ge 3$ and to fractal/Cayley graphs with polynomial volume growth, revealing universality in the percolation behavior of level sets and interlacements across diverse spaces and providing tools to study percolation thresholds in non-Euclidean environments.

Abstract

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like $\mathbb{Z}^d$, for $d \geqslant 3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.

Figures (2)

• Figure 1: A graph of the form $G_2=G'\times \mathbb{Z}$, with $G'$ the discrete Sierpinski gasket.
• Figure 2: The grey regions are not part of the graph, but there is always a (purple) path between $y$ and $y'$ in $U_0^{\mathsf{c}} \cap U_2$. In green, the balls $B(z_i,\varepsilon^2s)$ are included in $(A_1\cup A_2)^{\mathsf{c}}\subset B^{\mathsf{c}}$.

Theorems & Definitions (98)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Remark 2.2: Equivalence to heat kernel bounds
• Lemma 2.3
• proof
• Theorem 2.4
• Lemma 3.1
• proof
• Lemma 3.2