# Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

## TL;DR

This paper analyzes the geometry of the Gibbs measure for a scale-inhomogeneous 2D Gaussian free field, computing free energy, overlap distribution, and establishing ultrametricity and Ruelle cascade structure.

## Contribution

It extends the understanding of the Gaussian free field by computing the limiting free energy and overlap distribution, and proving ultrametricity and Ruelle cascade structure.

## Key findings

- Limiting free energy computed for the scale-inhomogeneous GFF.
- Limiting two-overlap distribution determined, matching GREM predictions.
- Extended Ghirlanda-Guerra identities hold, implying ultrametricity and Ruelle cascade law.

## Abstract

We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V_N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V_N) and the limiting two-overlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Generalized Random Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova (2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra identities hold exactly in the limit. As a corollary, the limiting array of overlaps is ultrametric and the limiting Gibbs measure has the same law as a Ruelle probability cascade.