For first passage percolation on with i.i.d. bounded edge weights, we consider the upper tail large deviation event, i.e., the rare situation where the first passage time between two points at distance n is macroscopically larger than typical. It was shown by Kesten [24] that the probability of this event decays as . However, the question of existence of the rate function, i.e., whether the log-probability normalized by n² tends to a limit, remains open. We show that under some additional mild regularity assumption on the passage time distribution, the rate function for upper tail large deviation indeed exists. The key intuition behind the proof is that a limiting metric structure that is atypical causes the upper tail large deviation event. The formal argument then relies on an approximate version of the above which allows us to use independent copies of the large deviation environment at a given scale to form an environment at a larger scale satisfying the large deviation event. Using this, we compare the upper tail probabilities for various values of n. © 2021 Wiley Periodicals LLC.

中文翻译：

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首道渗流中上尾大偏差

对于具有 iid 有界边缘权重的第一次通过渗透，我们考虑上尾大偏差事件，即距离为n 的两点之间的第一次通过时间在宏观上大于典型情况的罕见情况。Kesten [24] 表明该事件的概率衰减为。然而，率函数是否存在的问题，即对数概率是否被n ²归一化趋于极限，保持开放。我们表明，在对通过时间分布的一些额外的温和规律性假设下，上尾大偏差的速率函数确实存在。证明背后的关键直觉是，非典型的限制度量结构会导致上尾大偏差事件。然后，形式论证依赖于上述的近似版本，这允许我们使用给定规模的大偏差环境的独立副本来形成满足大偏差事件的更大规模的环境。使用这个，我们比较了n 的各种值的上尾概率。© 2021 威利期刊有限责任公司。