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Random sparse sampling in a Gibbs weighted tree

Let $μ$ be the geometric realization on $[0,1]$ of a Gibbs measure on $Σ=\{0,1\}^{\mathbb{N}}$ associated with a Hölder potential. The thermodynamic and multifractal properties of $μ$ 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 $\{I_w\}_{w\in Σ^*}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the set of finite dyadic words $Σ^*$. Fix $η\in(0,1)$, and a sequence $(p_w)_{w\in Σ^*}$ of independent Bernoulli variables of parameters $2^{-|w|(1-η)}$ ($|w|$ is the length of $w$). We consider the (very sparse) remaining values $\widetildeμ=\{μ(I_w): w\in Σ^*, p_w=1\}$. We prove that when $η<1/2$, it is possible to entirely reconstruct $μ$ from the sole knowledge of $\widetildeμ$, while it is not possible when $η>1/2$, hence a first phase transition phenomenon. We show that, for all $η\in (0,1)$, it is possible to reconstruct a large part of the initial multifractal structure of $μ$, via the fine study of $\widetildeμ$. 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.