Spherical Gaussian multiplicative chaos interaction data

Let $X$ be the centered random distribution on the Riemann sphere with covariance $K(z,w)=\log (1\mathrm{/}\mathrm{\mid }z-w\mathrm{\mid })+\log \mathrm{\mid }z{\mathrm{\mid }}_{+}+\log \mathrm{\mid }w{\mathrm{\mid }}_{+}$ and background measure $d{A}_0(z)=\mathrm{\mid }z{\mathrm{\mid }}_{+}^{-4}dxdy$, as specified by the earlier GFF definition. Let ${\rho }_{\epsilon }$ range over its smooth local averaging families. For $0<\gamma <2$, define the regularized positive random measures

$M_{\gamma ,\epsilon }(dz)=\exp \text{\hspace{0.17em}⁣}(\gamma X_{\epsilon }(z)-\frac{{\gamma }^2}{2}\mathbb{E}[X_{\epsilon }(z{)}^2])d{A}_0(z)\mathrm{.}$

A spherical Gaussian multiplicative chaos datum at parameter $\gamma$ is a finite positive random Radon measure $M_{\gamma }$ on $S$ such that for every $f\in C(S;\mathbb{R})$, ${\int }_{S}fd{M}_{\gamma ,\epsilon }$ converges in probability to ${\int }_{S}fd{M}_{\gamma }$ as $\epsilon \downarrow 0$, with the same limit for every admitted smoothing family. “Nonzero” means $\mathbb{P}(M_{\gamma }(S)>0)=1$. This definition specifies the required limit as data; it does not by itself assert that such data exist for every $0<\gamma <2$.

Gaussian free field on the Riemann sphere in the Liouville convention

Radon measure on an LCH space

The subcritical parameter range in the cited construction is $0<\gamma <2$ in real dimension two. The local $L^2$ argument on the assigned page covers only $0<\gamma <\sqrt{2}$; a complete local construction of the Radon-measure limit on the full interval remains a separate proof obligation.