论文标题
随机立方平面图收敛到布朗球体
Random cubic planar graphs converge to the Brownian sphere
论文作者
论文摘要
在本文中，随机连接的立方平面图的缩放极限（分别是多编码）被证明是布朗球体。 证明基本上是两个主要步骤。首先，由于已知的立方平面图分解为其三个连接的组件，因此，随机立方平面图的度量结构被其线性尺寸的独特3连接组件（具有修改的距离）很好地近似。 然后，惠特尼的定理确保了3个连接的立方平面图是简单的三角剖分的双重，众所周知，缩放极限是布朗尼球体。 Curien和Le Gall最近开发了一个框架，以研究一般三角形和双重距离的距离的修改。通过将此框架扩展到简单的三角剖分，可以表明，具有修改距离的3个相互连接的立方平面图与双重三角剖分共同收敛到布朗尼球体。
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.
