  The infinite discrete stable Boltzmann maps are "heavy-tailed"
generalisations of the well-known Uniform Infinite Planar Quadrangulation. Very
efficient tools to study these objects are Markovian step-by-step explorations
of the lattice called peeling processes. Such a process depends on an algorithm
which selects at each step the next edge where the exploration continues. We
prove here that, whatever this algorithm, a peeling process always reveals
about the same portion of the map, thus growing roughly metric balls. Applied
to well-designed algorithms, this easily enables us to compare distances in the
map and in its dual, as well as to control the so-called pioneer points of the
simple random walk, both on the map and on its dual.
