On intermediate level sets of two-dimensional discrete Gaussian Free Field

TL;DR

This paper analyzes the scaling limit of intermediate level sets of the 2D discrete Gaussian Free Field, revealing their connection to Liouville Quantum Gravity measures and detailed local structure at heights near the maximum.

Contribution

It provides a detailed description of the local structure and distribution of intermediate level sets of the DGFF, linking them to LQG measures and sharpening previous results.

Findings

Level set positions follow LQG measure distribution

Field values have exponential intensity measure

Size of level sets converges to LQG measure mass

Abstract

We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattice) versions of suitably regular continuum domains $D\subset\mathbb C$ and describe the scaling limit, including local structure, of the level sets at heights growing as a $\lambda$-multiple of the height of the absolute maximum, for any $\lambda\in(0,1)$. We prove that, in the scaling limit, the scaled spatial position of a typical point $x$ sampled from this level set is distributed according to a Liouville Quantum Gravity (LQG) measure in $D$ at parameter equal $\lambda$-times its critical value, the field value at $x$ has an exponential intensity measure and the configuration near $x$ reduced by the value at $x$ has the law of a pinned DGFF reduced by a suitable multiple of the potential kernel. In particular, the law of the total size of the level set, properly-normalized, converges that that of the total mass of the LQG measure. This sharpens considerably an earlier conclusion by Daviaud.