The hyperbolic Brownian plane

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 by Curien (Probab Theory Relat Fields 165(3):509–540, 2016). The law of the hyperbolic Brownian plane is obtained after biasing the law of the Brownian plane of Curien and Le Gall (J Theoret Probab 27(4):1249–1291, 2014) by an explicit martingale depending on its perimeter and volume processes studied by Curien and Le Gall (Probab Theory Relat Fields 166(1):187–231, 2016). 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.