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

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.

22 pages, accepted by Acta Mathematica Sinica, English Series