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 $2{\mu }^{\prime }$, where ${\mu }^{\prime }$ is the critical GMC measure.

15 pages, no figures needed