Boundaries of planar graphs, via circle packings
Boundaries of planar graphs, via circle packings
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平面图的边界，通过圆堆积
DOI：
10.1214/15-aop1014
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Asaf Nachmias
中科院分区：
文献类型：
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作者：
Omer Angel;M. Barlow;O. Gurel;Asaf Nachmias
给出了平面的瞬时有界度三角剖分的Poisson和Martin边界的几何表示，用单位圆盘中的圆填充表示。(对于Mobius变换，这种压缩是唯一的。)更确切地说，我们证明了图上的任何有界调和函数都是圆盘边界上某个可测函数的调和扩张，极值正调和函数的空间，即Martin边界，与单位圆是同胚的。我们的所有结果都适用于平面图的任何“好”嵌入，即嵌入在单位圆盘上的直线使得角度一致地远离$0$和$\pi$，并且相邻边的长度是可比较的。此外，我们还证明了在平面图的良好嵌入下，随机游动通过足够宽的圆弧离开圆盘的概率至少是一个常数，并且这类图上的布朗运动需要$r^2$才能离开半径为$r$的圆盘。这些回答了切尔卡克(2014)最近提出的一个问题。
We provide a geometric representation of the Poisson and Martin boundaries of a transient, bounded degree triangulation of the plane in terms of its circle packing in the unit disc. (This packing is unique up to Mobius transformations.) More precisely, we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk, and that the space of extremal positive harmonic functions, that is, the Martin boundary, is homeomorphic to the unit circle. All our results hold more generally for any "good"-embedding of planar graphs, that is, an embedding in the unit disc with straight lines such that angles are bounded away from $0$ and $\pi$ uniformly, and lengths of adjacent edges are comparable. Furthermore, we show that in a good embedding of a planar graph the probability that a random walk exits a disc through a sufficiently wide arc is at least a constant, and that Brownian motion on such graphs takes time of order $r^2$ to exit a disc of radius $r$. These answer a question recently posed by Chelkak (2014).