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}$[M_n]^k$andhencetoestablishexponentialmomentboundsfor$M_n$\mathrm{.}Thisimpliesthatthereexists$c>0$suchthatforall$n$andall$\lambda\geq0$, $$\mathbf{P}\{M_n>\lambda \mathbf{E}[M_n]\}\le 2e^{-c\lambda }\mathrm{.}$$ Using similar techniques, we then establish a second moment result for a specific conditioned random walk which enables us to prove that for any $\alpha <4\mathrm{/}5$, there exist $C$ and $c^{\mathrm{\prime }}>0$ such that for all $n$ and $\lambda >0$, $$\mathbf{P}\{M_n<{\lambda }^{-1}\mathbf{E}[M_n]\}\le C{e}^{-c^{\mathrm{\prime }}{\lambda }^{\alpha }}\mathrm{.}$$