The snake in the Brownian sphere
The Brownian sphere is a random metric space, homeomorphic to the
two-dimensional sphere, which arises as the universal scaling limit of many
types of random planar maps. The direct construction of the Brownian sphere is
via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection.
The CVS bijection maps labeled trees to planar maps, and the continuous version
maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to
the Brownian sphere. In this work, we describe the inverse of the continuous
CVS bijection, by constructing the Brownian snake as a measurable function of
the Brownian sphere. Special care is needed to work with the orientation of the
Brownian sphere.
67bd8759fdecc637bd621e6b
Omer Angel
67bd8759fdecc637bd621e6c
Emmanuel Jacob
67bd8759fdecc637bd621e6d
Brett Kolesnik
67bd8759fdecc637bd621e6e
Grégory Miermont
The snake in the Brownian sphere
The Brownian sphere is a random metric space, homeomorphic to the
two-dimensional sphere, which arises as the universal scaling limit of many
types of random planar maps. The direct construction of the Brownian sphere is
via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection.
The CVS bijection maps labeled trees to planar maps, and the continuous version
maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to
the Brownian sphere. In this work, we describe the inverse of the continuous
CVS bijection, by constructing the Brownian snake as a measurable function of
the Brownian sphere. Special care is needed to work with the orientation of the
Brownian sphere.
Brownian sphere random metric space Cori--Vauquelin--Schaeffer (CVS) bijection planar maps Aldous' continuum random tree Brownian labels Brownian snake measurable function orientation