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

- Date: 09/13/2006

Jason Schweinsberg (University of California at San Diego)

University of British Columbia

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.

Probability Seminar 2006