Infinite random planar maps related to Cauchy processes

Timothy Budd,Cyril Marzouk,Nicolas Curien

doi:10.5802/jep.82

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.

Highlights

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
Boltzmann planar maps with high degrees. — In all this work, we consider rooted planar maps, i.e., graphs embedded in the two-dimensional sphere or in the plane, and equipped with a distinguished oriented edge, the root-edge of the map, whose origin vertex is the root-vertex of the map, and the face adjacent to the right of the root-edge is called the root-face
In passing we prove several estimates on random walks converging towards an asymmetric Cauchy process

Summary

Intermezzo

The map M(∞), a half-plane version of M∞. — Let us briefly introduce another model of infinite random planar maps which we shall consider in Sections 4.2 and 5. The laws P( ) converge weakly for the local topology as → ∞; we shall denote by P(∞) the limit, which is a distribution on the set of one-ended rooted bipartite maps with a root-face of infinite degree. These are commonly referred to a maps of the half-plane. It is related to the process conditioned to survive (forever) as follows: let f ↑ be the density of Υ↑1, there exists a constant C1 > 0 such that f ↑(x) = C1x1/2f (x).

First passage percolation distance

Graph distance

The upper bound

Bernoulli percolation