布朗仙人掌离散仙人掌的尺度极限
The Brownian Cactus I. Scaling limits of discrete cactuses
摘要：
一个点图的仙人掌是与这个图相关联的一棵离散树。类似地，对于每一个点测地度量空间$E$，我们可以将一个称为$E$的连续仙人掌的$\R$ -树联系起来。我们证明在一般假设下，仙人掌随机平面地图分布根据玻尔兹曼权重和条件有一个固定的大量的顶点收敛分布到一个极限空间称为布朗仙人掌，在Gromov-Hausdorff意义上。此外，布朗仙人掌可以解释为所谓布朗映射的连续仙人掌。
英文摘要：
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.
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