Thick points of the planar GFF are totally disconnected for all $\gamma\ne 0$

TL;DR

This paper proves that the set of gamma-thick points in a planar Gaussian free field is almost surely totally disconnected for all nonzero gamma, using couplings with nested CLE4 and implications for supercritical LQG metrics.

Contribution

It establishes the total disconnectedness of gamma-thick points for all gamma not equal to zero in planar GFFs, linking it to CLE4 nesting fields.

Findings

Gamma-thick points are totally disconnected for all gamma ≠ 0.

The set of singular points in supercritical LQG metrics is totally disconnected.

Thick points coincide with those of the weighted CLE4 nesting field.

Abstract

We prove that the set of $\gamma$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $\gamma \neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_{\kappa}$, $\kappa \in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.