Let
$\unicode[STIX]{x1D707}$
be the projection on
$[0,1]$
of a Gibbs measure on
$\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$
(or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of
$\unicode[STIX]{x1D707}$
are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let
$\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$
stand for the collection of dyadic subintervals of
$[0,1]$
naturally indexed by the finite dyadic words. Fix
$\unicode[STIX]{x1D702}\in (0,1)$
, and a sequence
$(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$
of independent Bernoulli variables of parameters
$2^{-|w|(1-\unicode[STIX]{x1D702})}$
. We consider the (very sparse) remaining values
$\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$
. We study the geometric and statistical information associated with
$\widetilde{\unicode[STIX]{x1D707}}$
, and the relation between
$\widetilde{\unicode[STIX]{x1D707}}$
and
$\unicode[STIX]{x1D707}$
. To do so, we construct a random capacity
$\mathsf{M}_{\unicode[STIX]{x1D707}}$
from
$\widetilde{\unicode[STIX]{x1D707}}$
. This new object fulfills the multifractal formalism, and its free energy is closely related to that of
$\unicode[STIX]{x1D707}$
. Moreover, the free energy of
$\mathsf{M}_{\unicode[STIX]{x1D707}}$
generically exhibits one first order and one second order phase transition, while that of
$\unicode[STIX]{x1D707}$
is analytic. The geometry of
$\mathsf{M}_{\unicode[STIX]{x1D707}}$
is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct
$\unicode[STIX]{x1D707}$
from
$\widetilde{\unicode[STIX]{x1D707}}$
by using the almost multiplicativity of
$\unicode[STIX]{x1D707}$
and concatenation of words is discussed as well.