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November 2010 Exponential tail bounds for loop-erased random walk in two dimensions
Martin T. Barlow, Robert Masson
Ann. Probab. 38(6): 2379-2417 (November 2010). DOI: 10.1214/10-AOP539


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.


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Martin T. Barlow. Robert Masson. "Exponential tail bounds for loop-erased random walk in two dimensions." Ann. Probab. 38 (6) 2379 - 2417, November 2010.


Published: November 2010
First available in Project Euclid: 24 September 2010

zbMATH: 1207.60035
MathSciNet: MR2683633
Digital Object Identifier: 10.1214/10-AOP539

Primary: 60G50 , 60J65

Keywords: exponential tail bounds , growth exponent , Loop-erased random walk

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 6 • November 2010
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