Let µ be the geometric realization on [0, 1] of a Gibbs measure on Σ = {0, 1} 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 {Iw}w∈Σ * stand for the collection of dyadic subintervals of [0, 1] naturally indexed by the set of finite dyadic words Σ *. Fix η ∈ (0, 1), and a sequence (pw)w∈Σ * of independent Bernoulli variables of parameters 2 −|w|(1−η) (|w| is the length of w). We consider the (very sparse) remaining values µ = {µ(Iw) : w ∈ Σ * , pw = 1}. We prove that when η < 1/2, it is possible to entirely reconstruct µ from the sole knowledge of µ, while it is not possible when η > 1/2, hence a first phase transition phenomenon. We show that, for all η ∈ (0, 1), it is possible to reconstruct a large part of the initial multifractal structure of µ, via the fine study of µ. 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.