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 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 loop-erased 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,..., E[M(n)(k)] <= C(k)k!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 lambda >= 0, P{M(n) > lambda E[M(n)]} <= 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 alpha < 4/5, there exist C and c' > 0 such that for all n and lambda > 0, P{M(n) < lambda(-1) E[M(n)]} <= Ce(-c'lambda alpha)