Brownian disks and the Brownian snake

Abstract

We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications of our construction. In particular, we prove that the uniform measure on the boundary can be obtained as the limit of the suitably normalized volume measure on a small tubular neighborhood of the boundary. We also prove that connected components of the complement of the Brownian net are Brownian disks, as it was suggested in the recent work of Miller and Sheffield. Finally, we show that connected components of the complement of balls centered at the distinguished point of the Brownian map are independent Brownian disks, conditionally on their volumes and perimeters.