EXPONENTIAL TAIL BOUNDS FOR LOOP-ERASED RANDOM WALK IN TWO DIMENSIONS

Let M n be the number of steps of the loop-erasure of a simple random walk on ${\Bbb Z}^{2}$ from the origin to the circle of radius n. We relate the moments of M n to Es(n), the probability that a random walk and an independent looperased random walk both started at the origin do not intersect up to leaving the ball of radius n. This allows us to show that there exists C such that for all n and all k = 1, 2,..., ${\bf E}[M_{n}^{k}]\leq C^{k}k!{\bf E}[M_{n}]^{k}$ and hence to establish exponential moment bounds for M n . This implies that there exists c > 0 such that for all n and all λ ≥ 0, ${\bf P}\{M_{n}>\lambda {\bf E}[M_{n}]\}\leq 2e^{-c\lambda}$ . Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any α < 4/5, there exist C and c' > 0 such that for all n and λ > 0, ${\bf P}\{M_{n}