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.
Highlights
Gaussian multiplicative chaos (GMC) theory aims to give a meaning to the heuristic volume form “eΓd Leb”, where Γ is some rough Gaussian field that is not defined pointwise
The theory was developed for a larger class of Gaussian fields, and named as Gaussian multiplicative chaos, by Kahane [Kah85]
GMC measures corresponding to the 2D continuum Gaussian free field (GFF), have recently become an active area of study, due to their links to the probabilistic description of 2D Liouville quantum gravity [DS11, DKRV16]
Summary
Gaussian multiplicative chaos (GMC) theory aims to give a meaning to the heuristic volume form “eΓd Leb”, where Γ is some rough Gaussian field that is not defined pointwise. Such constructions first appeared for Gaussian free fields in the early 70s [HK71], where. We study how the Liouville measures μγ vary, for a fixed underlying field, when the parameter γ tends to the critical parameter γ = 2 from below. We strongly use the results on the Seneta-Heyde scaling of the Liouville measure proved in [APS17]. The rest of the article is structured as follows: we start with basic definitions followed by a few preliminary lemmas; in Section 4 we prove the main result; and we discuss some extensions
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