Random sparse sampling in a Gibbs weighted tree

Let $\mu$ be the geometric realization on $[0,1]$ of a Gibbs measure on $\Sigma=\{0,1\}^{\mathbb{N}}$ associated with a H\"older potential. The thermodynamic and multifractal properties of $\mu$ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied. More precisely, let $\{I_w\}_{w\in \Sigma^*}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the set of finite dyadic words $\Sigma^*$. Fix $\eta\in(0,1)$, and a sequence $(p_w)_{w\in \Sigma^*}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\eta)}$ ($|w|$ is the length of $w$). We consider the (very sparse) remaining values $\widetilde\mu=\{\mu(I_w): w\in \Sigma^*, p_w=1\}$. We prove that when $\eta<1/2$, it is possible to entirely reconstruct $\mu$ from the sole knowledge of $\widetilde\mu$, while it is not possible when $\eta>1/2$, hence a first phase transition phenomenon. We show that, for all $\eta \in (0,1)$, it is possible to reconstruct a large part of the initial multifractal structure of $\mu$, via the fine study of $\widetilde\mu$. After reorganization, these coefficients give rise to a random capacity with new remarkable scaling and multifractal properties: its $L^q$-spectrum exhibits two phase transitions, and has a rich thermodynamic and geometric structure.