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?