Let Mn 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 Mn 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[Mnk] ≤ Ckk!E[Mn]k and hence to establish exponential moment bounds for Mn. This implies that there exists c > 0 such that for all n and all λ ≥ 0,
P{Mn > λE[Mn]} ≤ 2e−cλ.
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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{Mn < λ−1E[Mn]} ≤ Ce−c'λα.