Critical Liouville measure as a limit of subcritical measures

TL;DR

This paper investigates the behavior of Gaussian multiplicative chaos measures near the critical parameter, demonstrating convergence of scaled subcritical measures to a critical measure in probability.

Contribution

It establishes the limiting behavior of subcritical GMC measures as the parameter approaches criticality, revealing a precise convergence result.

Findings

Scaled subcritical measures converge to twice the critical measure

The convergence occurs in probability as the parameter approaches criticality

Provides a rigorous link between subcritical and critical GMC measures

Abstract

We study how the Gaussian multiplicative chaos (GMC) measures $\mu^\gamma$ corresponding to the 2D Gaussian free field change when $\gamma$ approaches the critical parameter $2$. In particular, we show that as $\gamma\to 2^{-}$, $(2-\gamma)^{-1}\mu^\gamma$ converges in probability to $2\mu'$, where $\mu'$ is the critical GMC measure.