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 ${\mathbb{Z}}^2$ 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 $k=1,2,...,\mathbf{E}[M_n^k]\le C^{k}k!\mathbf{E}[M_n{]}^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and all $\lambda \ge 0$, \[\mathbf{P}\{M_n>λ\mathbf{E}[M_n]\}\leq2e^{-cλ}.\] Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $\alpha <4/5$, there exist $C$ and $c^{\prime }>0$ such that for all $n$ and $\lambda >0$, \[\mathbf{P}\{M_n<λ^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'λ^α}.\]

Published in at http://dx.doi.org/10.1214/10-AOP539 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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