Critical Liouville measure as a limit of subcritical measures

[摘要] 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.

[学科分类] 统计和概率

[关键词] Multiplicative chaos;Liouville measure;Liouville quantum gravity;multiplicative cascades;critical GMC measure;first passage sets;Gaussian free field;local sets