Critical Liouville measure as a limit of subcritical measures

Electronic Communications in Probability

Juhan Aru¹ 

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

DOI  :  10.1214/19-ECP209

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 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.

【 授权许可】

CC BY