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 ${\mathrm{e}}^{\sqrt{r}}$. 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.

36 pages, 8 figures. Comments are welcome

References in corpus (1)

• A Boltzmann approach to percolation on random triangulations

Cited by in corpus (4)

• The trunks of CLE(4) explorations
• Simple peeling of planar maps with application to site percolation
• The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model
• Distances on the ${\mathrm{CLE}}_4$, critical Liouville quantum gravity and $3/2$-stable maps