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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 $M_n$. 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Barlow, Robert Masson","submitted_at":"2009-10-27T00:30:16Z","abstract_excerpt":"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 $M_n$. 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