# Infinite random planar maps related to Cauchy processes

### Abstract

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 (Scaling limits of random planar maps with large faces, Ann. Probab. 39, 1 (2011), 1-69) studied in (Budd & Curien, Geometry of infinite planar maps with high degrees, Electron. J. Probab. (to appear)) 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 ball of radius $r$ for the graph distance has an intermediate rate of growth and scales as $e^{\sqrt{r}}$. We also perform first passage percolation with exponential edge-weights and show that the volume growth for the fpp-distance scales as $e^r$. 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.

Type
Publication
Journal de l’École Polytechnique - Mathématiques, 5 (2018), p. 749-791
Date