proof of Proposition \[Proposition lower envelope A(t)\] provides an explicit $q_0$, with a rather complicated expression given in Remark \[Remark exponent lower envelope\] in Section 5. We do not claim however that the bounds of the two above Propositions are optimal.

In Section \[Section Application to random maps\], we shall recall the connection between growth-fragmentations and a family of random surfaces, and apply the above results to the latter, refining in particular some results for the Brownian map.

Area near the origin {#Section area near the origin}
====================

We assume in this section that and hold. Our goal is to prove Theorem \[Theorem area GF\]. We first look at the expectation of $A(\epsilon)$ as $\epsilon\to 0+$.

Behaviour of the expectation
----------------------------

Under $\mathcal{P}_1^-$, the spine is chosen according to $\mathcal{A}$. We can hence study $A$ through the lifetime of the spine $I$, thanks to the following relation.

\[Lemma expect A(epsilon) = distr function of I\] For all $t\geq 0$, it holds that $$\begin{aligned}
            \mathcal{E}_1(A(t))=\mathbb{P}_1^-(I\leq t).
        \end{aligned}$$

This is a straightforward consequence of Lemmas 2.1 and 2.2 in [@G18]. Thanks to Lemma \[Lemma expect A(epsilon) = distr function of I\], we can deduce

\[Lemma asymp expect A(espilon)\] It holds that that 