For first passage percolation on @math 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 @math , is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as @math . However the question of existence of the rate function i.e., whether the log-probability normalized by @math tends to a limit, had remained 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. Our proof can be generalized to work in higher dimensions and for the corresponding problem in last passage percolation as well. The key intuition behind the proof is that a limiting metric structure which 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 dilate the large deviation environment to compare the upper tail probabilities for various values of @math
Kesten @cite_18 considered both upper and lower tail large deviations for first passage percolation. Let @math (throughout this section for brevity we will use @math to denote the passage time between @math and @math in @math although it was initially defined only for @math ) denote the lower tail large deviation event. Using a subadditive argument, Kesten showed that for @math ,
@cite_18
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