## 2009 Probability Seminar - 06

- Date: 03/11/2009

University of British Columbia

Second moment estimates for the growth exponent of loop-erased random walk

The loop-erased random walk Y^n is the process obtained by running a

random walk in Z^d from the origin to the first exit time of the ball

of radius n and then chronologically erasing its loops. If we let X_n

denote the number of steps of Y^n then the growth exponent a is defined

to be such that E[X_n] grows like n^a. The value of a (or even its

existence) depends on the dimension d. In this talk I'll focus on d=2

where it's been shown that a = 5/4. What we want to know is how close

is X_n to its mean? By the Markov inequality one gets that P(X_n >

bE[X_n]) < b^{-1}. The goal of this talk will be to show a similar

lower bound: P(X_n < bE[X_n]) < b^{c} for some c>0.

3:00pm-4:00pm, WMAX 216