Exponential tail bounds for loop-erased random walk in two dimensions
Exponential tail bounds for loop-erased random walk in two dimensions
复制标题
二维循环擦除随机游走的指数尾界
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Robert Masson
中科院分区：
文献类型：
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作者：
M. Barlow;Robert Masson
让mn的步骤的一个简单的随机漫步loop-erasureℤ2从起源到圆的半径n。我们的时刻与M n (n),随机游走的概率和独立loop-erased随机游走都开始在原点不相交离开球的半径n。这使我们能够证明存在C, n和k = 1, 2,…， E[M k n]≤，从而建立mn的指数矩界。这意味着存在c > 0，使得对于所有n和所有λ≥0，P{M n > λ e [M n]}≤2e -cλ。使用类似的技术，我们建立了一个特定条件随机漫步的第二矩结果，使我们能够证明对于任意α 0，使得对于所有n和λ >, P{M n < λ -1 E[M n]}≤Ce -c 'λα。
Let M n be the number of steps of the loop-erasure of a simple random walk on 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, ..., E[M k n ]≤ 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 ≥ 0, P{M n > λE[M n ]} ≤ 2e ―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 0 such that for all n and λ > 0, P{M n < λ ―1 E[M n ]} ≤ Ce ―c'λα .