Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.
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机译：假设$ G_j $是有限连接平面图的序列，并且在每个$ G_j $中随机地均匀选择一个称为根的特殊顶点。我们介绍了这种图的分布极限$ G $的概念。假定$ G_j $中顶点的顶点度是有界的，并且该界不依赖于$ j $。然后，在传递到子序列后，该限制存在，并且是一个随机根图$ G $。我们证明，一个$ G $经常出现。证明涉及圆包装定理。这项工作的动机来自于随机球面三角剖分理论。
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