## The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus

• Date: 09/13/2006
Lecturer(s):

Jason Schweinsberg (University of California at San Diego)

Location:

University of British Columbia

Topic:

Let x and y be points chosen uniformly at random from the
four-dimensional discrete torus with side length n. We show that the
length of the loop-erased random walk from x to y is of order n2 (log
n)^{1/6}, resolving a conjecture of Benjamini and Kozma. We also show
that the scaling limit of the uniform spanning tree on the
four-dimensional discrete torus is the Brownian continuum random tree
of Aldous. Our proofs use the techniques developed by Peres and
Revelle, who studied the scaling limits of the uniform spanning tree on
a large class of finite graphs that includes the d-dimensional discrete
torus for d >= 5, in combination with results of Lawler concerning
intersections of four-dimensional random walks.

Other Information:

Probability Seminar 2006