Exponential tail bounds for loop-erased random walk in two dimensions

Abstract

Let M_n be the number of steps of the loop-erasure of a simple random walk on ℤ² 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^kk!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,

P{M_n > λE[M_n]} ≤ 2e^−cλ.

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,

P{M_n < λ⁻¹E[M_n]} ≤ Ce^−c'λα.