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Recurrence of planar graph limits

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Random Geometry

Co-author: Asaf Nacmias (Tel Aviv University)

What does a random planar triangulation on n vertices looks like? More precisely, what does the local neighbourhood of a fixed vertex in such a triangulation looks like? When n goes to infinity, the resulting object is a random rooted graph called the Uniform Infinite Planar Triangulation (UIPT). Angel, Benjamini and Schramm conjectured that the UIPT and similar objects are recurrent, that is, a simple random walk on the UIPT returns to its starting vertex almost surely. In a joint work with Asaf Nachmias, we prove this conjecture. The proof uses the electrical network theory of random walks and the celebrated Koebe-Andreev-Thurston circle packing theorem. We will give an outline of the proof and explain the connection between the circle packing of a graph and the behaviour of a random walk on that graph.

This talk is part of the Isaac Newton Institute Seminar Series series.

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