The intermediate level-sets of the four-dimensional membrane model

TL;DR

This paper establishes the scaling limit of intermediate level-sets in a four-dimensional discrete membrane model, showing they converge in law to a sub-critical Gaussian multiplicative chaos measure of the continuum membrane model.

Contribution

It proves the existence of the scaling limit of intermediate level-sets and identifies it with a sub-critical GMC measure, advancing understanding of membrane models in four dimensions.

Findings

Confirmed the existence of the scaling limit of level-sets.

Identified the limit as a sub-critical Gaussian multiplicative chaos measure.

Established the law equivalence between discrete and continuum models.

Abstract

In this paper, we consider the discrete membrane model in four dimensions. We confirm the existence of the scaling limit of the intermediate (i.e., a multiple of the expected maximum) level-sets of the model, and show that it is equal in law to the sub-critical Gaussian multiplicative chaos (GMC) measure of the continuum membrane model.