The hyperbolic Brownian plane

TL;DR

This paper introduces the hyperbolic Brownian plane, a new random surface that is a near-critical scaling limit of hyperbolic triangulations, exhibiting exponential volume growth unlike the Brownian plane.

Contribution

It defines and analyzes the hyperbolic Brownian plane, extending the understanding of random surfaces with hyperbolic geometry and their scaling limits.

Findings

Exponential volume growth of the hyperbolic Brownian plane.

Law obtained via biasing the Brownian plane law with an explicit martingale.

Local properties similar to the Brownian plane, but different large-scale structure.

Abstract

We introduce and study a new random surface which we call the hyperbolic Brownian plane and which is the near-critical scaling limit of the hyperbolic triangulations constructed in arXiv:1401.3297. The law of the hyperbolic Brownian plane is obtained after biasing the law of the Brownian plane arXiv:1204.5921 by an explicit martingale depending on its perimeter and volume processes studied in arXiv:1409.4026. Although the hyperbolic Brownian plane has the same local properties as those of the Brownian plane, its large scale structure is much different since we prove e.g. that is has exponential volume growth.