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The uniform spanning forest of planar graphs

Presented by: 
A Nachmias Tel Aviv University
Date: 
Thursday 23rd April 2015 - 09:00 to 10:00
Venue: 
INI Seminar Room 1
Abstract: 
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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons