Bipolar oriented random planar maps with large faces and exotic SLE$_\kappa(\rho)$ processes

TL;DR

This paper studies bipolar oriented random planar maps with heavy-tailed faces, showing their convergence to a limit described by correlated stable Lévy processes and connecting this to SLE$__( ho)$ processes on Liouville quantum gravity.

Contribution

It establishes the scaling limit of bipolar oriented maps with heavy-tailed faces as an SLE$__( ho)$ process on Liouville quantum gravity, linking stable Lévy processes to these maps.

Findings

Convergence of contour functions to correlated stable Lévy processes.

Identification of the map's scaling limit with SLE$__( ho)$ processes.

Infinite volume limit in the Benjamini-Schramm topology.

Abstract

We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $\alpha \in (1,2)$ that if the face degree is in the domain of attraction of an $\alpha$-stable L\'evy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $\alpha$-stable L\'evy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE$_\kappa(\rho)$ process with $\rho = \kappa-4 < -2$ on $\sqrt{\kappa}$-Liouville quantum gravity for $\kappa \in (4/3,2)$ where $\alpha, \kappa$ are related by $\alpha = 4/\kappa-1$.