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

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 $k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and all $λ\geq0$, $\mathbf{P}\{M_n>λ\mathbf{E}[M_n]\}\leq2e^{-cλ}.$ Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $α<4/5$, there exist $C$ and $c'>0$ such that for all $n$ and $λ>0$, $\mathbf{P}\{M_n<λ^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'λ^α}.$

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