Brownian disks and the Brownian snake
Brownian disks and the Brownian snake
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DOI：
10.1214/18-aihp882
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发表时间：
2017-04
期刊：
影响因子：
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通讯作者：
J. Gall
中科院分区：
文献类型：
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作者：
J. Gall
我们提供了一个新的布朗圆盘，这已被定义为Bettinelli和Miermont的四边形与边界的尺度限制时，边界大小趋于无穷大的建设。我们的方法非常类似于布朗映射的构造，但它利用了最近引入的布朗蛇的正行程测度。这个偏移测量涉及一个随机连续树，其顶点被分配非负标签，这对应于我们的方法中的布朗盘从边界的距离。我们提供了几个应用我们的建设。特别地，我们证明了边界上的一致测度可以作为边界的小管邻域上的适当归一化体积测度的极限而得到。我们还证明了布朗网的补的连通分量是布朗圆盘，正如米勒和谢菲尔德最近的工作所建议的那样。最后，我们证明了以布朗映射的特征点为中心的球的补的连通分量是独立的布朗圆盘，条件是它们的体积和周长。
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.