Markovian explorations of random planar maps are roundish

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 graph called peeling processes. Such a process depends on an algorithm that at each step selects the next edge where the exploration continues. We prove here that, whatever the algorithm, a peeling process always reveals about the same portion of the map, thus growing roughly like metric balls. Applied to well-designed algorithms, this enables us to easily 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.