Professor Jason Schweinsberg




Tuesday, February 14, 2006 - 11:00pm


MSTB 254

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 n^2 (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.