We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.

中文翻译：

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平面图的位点渗透和等距不等式

我们将等渗不等式与新技术结合使用，以证明在给定的最小度条件下平面图的站点渗透阈值的上限。在这个过程中，我们证明了某些双曲图类的紧新的等距界线。这就为所有三角形和正方形双曲晶格建立了顶点等静常数，回答了里昂和佩雷斯的问题。我们证明了最低度至少为7的平面图的站点渗漏阈值与Benjamini和Schramm猜想的1/2相距不远，并且在Angel，Benjamini和Horesh的猜想中，临界概率最大为对于最小度数为6的平面三角剖分，则为1/2。对于更强的最小度数条件以及没有三角形面的图形，我们证明了附加的边界。