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Upper Tail Large Deviations in First Passage Percolation
Author(s) -
Basu Riddhipratim,
Ganguly Shirshendu,
Sly Allan
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.22010
Subject(s) - rate function , mathematics , bounded function , large deviations theory , event (particle physics) , upper and lower bounds , intuition , combinatorics , statistical physics , statistics , mathematical analysis , physics , philosophy , epistemology , quantum mechanics
For first passage percolation on ℤ 2 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 exp − Θ n 2. However, the question of existence of the rate function, i.e., whether the log‐probability normalized by n 2 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.