Pacific Institute for the Mathematical Sciences - PIMS

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