  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
$\lambda\geq0$, \[\mathbf{P}\{M_n>\lambda\mathbf{E}[M_n]\}\leq2e^{-c\lambda}.\]
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 $c'>0$ such that for all $n$ and $\lambda>0$,
\[\mathbf{P}\{M_n<\lambda^{-1}\mathbf{E}[M_n]\}\leq Ce^{-c'\lambda
^{\alpha}}.\]
