Infinite random planar maps related to Cauchy processes

We study the geometry of infinite random Boltzmann planar maps having weight of polynomial decay of order k−² for each vertex of degree k. These correspond to the dual of the discrete “stable maps” of Le Gall and Miermont [26] studied in [12] related to a symmetric Cauchy process, or alternatively to the maps obtained after taking the gasket of a critical O(2)loop model on a random planar map. We show that these maps have a striking and uncommon geometry. In particular we prove that the volume of the (hull of the) ball of radius r for the graph distance has an intermediate rate of growth and scales roughly as e^c√r. We also perform first passage percolation with exponential edge-weights and show that the volume growth for the fpp-distance scales as e^cr. Finally we consider site percolation on these lattices: although percolation occurs only at p = 1, we identify a phase transition at p = 1/2 for the length of interfaces. On the way we also prove new estimates on random walks attracted to an asymmetric Cauchy process.