The boundary of a square tiling of a graph coincides with the Poisson boundary
The boundary of a square tiling of a graph coincides with the Poisson boundary
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图的正方形平铺的边界与泊松边界重合
DOI：
10.1007/s00222-015-0601-0
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发表时间：
2013
影响因子：
3.1
通讯作者：
Agelos Georgakopoulos
中科院分区：
文献类型：
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作者：
Agelos Georgakopoulos
在回答Benjamini和Schramm(Ann Probab 24(3)：1219-1238,1996)的一个问题时，我们证明了具有有界度的唯一吸收(例如一端和瞬变)的平面图的Poisson边界可以几何地实现为圆，这源于Riemann映射定理的一个离散版本。这暗示了NorthShield的猜想(潜在肛门2(4)：299-314,1993)。我们的一些技术适用于非平面情况，可能会有更多的应用。当图也是双曲图时，在较温和的条件下，我们证明了几种边界结构的等价性。
Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219–1238, 1996), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann’s mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299–314, 1993). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions.