2009 Probability Seminar - 06

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.