# Infinite random planar maps related to Cauchy processes

## TL;DR

This paper investigates the unique geometric properties of infinite random planar maps associated with Cauchy processes, revealing intermediate and exponential growth rates in different metrics and identifying phase transitions in percolation.

## Contribution

It introduces a new class of infinite planar maps linked to Cauchy processes and analyzes their geometric and percolation properties, including growth rates and phase transitions.

## Key findings

- Volume of balls grows as e^{√r} in graph distance
- FPP volume growth scales as e^{r}
- Percolation exhibits a phase transition at p=1/2

## 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 $\mathrm{e}^{\sqrt{r}}$. We also perform first passage percolation with exponential edge-weights and show that the volume growth for the fpp-distance scales as $\mathrm{e}^{r}$. 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.