Abstract We consider the last passage percolation on the complete graph. Let G n = ( [ n ] , E n ) be the complete graph on vertex set [ n ] = { 1 , 2 , … , n } , and i.i.d. sequence { X e : e ∈ E n } be the passage times of edges. Denote by W n the largest passage time among all self-avoiding paths from 1 to n . First, it is proved that W n ∕ n converges to constant μ , where μ is called the time constant and coincides with the essential supremum of X e . Second, when μ = ∞ , lower and upper bounds for W n ∕ n are given, in particular, for a large class of passage times with light tails, a weak law of large number for W n is obtained. Finally, when μ ∞ , it is proved that the deviation probability P ( W n ∕ n ≤ μ − x ) decays as fast as e − Θ ( n 2 ) .

中文翻译：

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完整图上的最后一段渗透

摘要 我们考虑完整图上的最后一段渗透。令 G n = ( [ n ] , E n ) 是顶点集 [ n ] = { 1 , 2 , … , n } 上的完整图，而 iid 序列 { X e : e ∈ E n } 是边缘。用 W n 表示从 1 到 n 的所有自回避路径中的最大通过时间。首先，证明了 W n ∕ n 收敛于常数 μ ，其中 μ 称为时间常数，与 X e 的本质上重合。其次，当 μ = ∞ 时，给出了 W n ∕ n 的下界和上限，特别是对于具有轻尾的大类通过时间，得到了 W n 的弱大数定律。最后，当 μ ∞ 时，证明了偏差概率 P ( W n ∕ n ≤ μ − x ) 的衰减速度与 e − Θ ( n 2 ) 一样快。