Markovian explorations of random planar maps are roundish

Nicolas Curien (1) , Cyril Marzouk (2, 3)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Domaines: Probabilités [math.PR]

Nicolas Curien, Cyril Marzouk. Markovian explorations of random planar maps are roundish. Bulletin de la société mathématique de France, 2020, 148 (4), pp.709-732. ⟨10.24033/bsmf.2821⟩. ⟨hal-02052950⟩