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The uniform spanning forest of planar graphs

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Random Geometry

The free uniform spanning forest (FUSF) of an infinite graph G is obtained as the weak limit of the law of a uniform spanning tree on G_n, where G_n is a finite exhaustion of G. It is easy to see that the FUSF is supported on spanning graphs of G with no cycles, but it need not be connected. Indeed, a classical result of Pemantle (‘91) asserts that when G=Z^d, the FUSF is almost surely a connected tree if and only if d=1,2,3,4.

In this talk we will show that if G is a plane graph with bounded degrees, then the FUSF is almost surely connected, answering a question of Benjamini, Lyons, Peres and Schramm (‘01). An essential part of the proof is the circle packing theorem.

Joint work with Tom Hutchcroft.

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

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