Gaussian free field from Liouville quantum gravity?

Question

If $\Sigma$ is a Riemann surface, there are two measures on $\text{H}^s(\Sigma)$:

1. the Gaussian free field $h(z)$ and
2. the Gaussian multiplicative chaos $\mu(z)= \lim_{\epsilon\to0} e^{\gamma h_\epsilon(z)}\epsilon^{\gamma^2/2}$.

See for instance Theorem 2.1 of Berestycki's notes.

My question is: what is the relation between 1 and 2? For instance, is there a way of making a formula a bit like $\frac{d}{d\gamma}\mu(z)\vert_{\gamma=0}=h(z)$ precise?

Answer 1

Since $\mu(z)$ is defined in terms of $h(z)$, the relation in one direction should be clear. For the other direction, one can retrieve the GFF from the GMC volumes of balls via Theorem 1.1 of

Berestycki, Nathanaël; Sheffield, Scott; Sun, Xin, Equivalence of Liouville measure and Gaussian free field, Ann. Inst. Henri Poincaré, Probab. Stat. 59, No. 2, 795-816 (2023). arXiv:1410.5407.

In the case $h$ is centered, there result amounts to the convergence of $$ \frac{\log\mu(B_z(\epsilon)) - \mathbb{E}\log\mu(B_z(\epsilon))}{\gamma} $$ in probability to $h$ as $\epsilon\to 0$, where $B_z(\epsilon)$ is the ball of radius $\epsilon$ around $z$.