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.
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