The hyperbolic Brownian plane

Understand

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.