The Annals of Probability

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

Martin T. Barlow and Robert Masson

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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.

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]} ≤ Cec'λα.

Article information

Ann. Probab. Volume 38, Number 6 (2010), 2379-2417.

First available in Project Euclid: 24 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60J65: Brownian motion [See also 58J65]

Loop-erased random walk growth exponent exponential tail bounds


Barlow, Martin T.; Masson, Robert. Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab. 38 (2010), no. 6, 2379--2417. doi:10.1214/10-AOP539.

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