Recurrence of Distributional Limits of Finite Planar Graphs

@article{Benjamini2000RecurrenceOD,
  title={Recurrence of Distributional Limits of Finite Planar Graphs},
  author={Itai Benjamini and Oded Schramm},
  journal={Electronic Journal of Probability},
  year={2000},
  volume={6},
  pages={1-13}
}
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… Expand

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