Percolations on random maps I: Half-plane models

Angel, Omer; Curien, Nicolas

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 405-431.

Nous étudions différentes percolations de Bernoulli sur les cartes aléatoires du demi-plan obtenues comme limites locales de triangulations ou quadrangulations planaires uniformes. En utilisant la propriété de Markov spatiale – ou épluchage (Geom. Funct. Anal. 13 (2003) 935–974) – de ces réseaux, nous prouvons une formule simple et universelle pour le paramètre critique de percolation par arêtes ou par sites sur ces cartes. Nos techniques nous permettent également de calculer certains exposants « annealed » presque-critiques et critiques comme la probabilité qu’un cluster ait un grand volume ou un grand périmètre.

We study Bernoulli percolations on random maps in the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process (Geom. Funct. Anal. 13 (2003) 935–974) of these random maps we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical annealed exponents related to percolation clusters such as the probabilities of a cluster having a large volume or perimeter.

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