The supercritical Gaussian multiplicative chaos renormalization conjecture

Let $X$ be the log-correlated Gaussian field defining the Gaussian multiplicative chaos measures $M^{\gamma,0}_\varepsilon$, and let $M'$ denote the derivative martingale. Assume $\gamma>\sqrt{2d}$ and set $\alpha=\frac{\sqrt{2d}}{\gamma}$. Let $N_\alpha$ be an independently scattered random measure whose conditional law given $M'$ is characterized by

$$\forall A\in\mathcal{B}(\mathbb{R}^d),\ \forall qgeqslant 0,\quad \mathds{E}[e^{-qN_\alpha(A)}\mid M']=e^{-q^\alpha M'(A)}.$$

Supercritical renormalization conjecture. As $\varepsilon\to0$,

$$(-\lnvarepsilon)^{\frac{3\gamma}{2\sqrt{2d}}}\varepsilon^{\gamma\sqrt{2d}-d}M^{\gamma,0}_\varepsilon(\mathrm{d}x)\stackrel{\mathrm{law}}{\longrightarrow}c_\gamma N_\alpha(\mathrm{d}x),$$

where $c_\gamma$ is a positive constant depending on $\gamma$.

The conjecture concerns universality of the supercritical renormalization across cutoff approximations. The source reports that it was proved for compactly supported covariance kernels and for specific cutoff schemes for the massless Gaussian free field in a bounded domain and the massive planar Gaussian free field, while convergence for a broad class of cutoff approximations remains open.