Critical Liouville measure as a limit of subcritical 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.