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
For the upper tail large deviations, Kesten showed that The existence of the limit was left open and this open question was re-iterated in @cite_10 (See Question 18), which we answer in our Theorem .
@cite_10
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We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.