Upper Tail Large Deviations in First Passage Percolation
Abstract
For first passage percolation on $\mathbb{Z}^2$ 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 (1986) that the probability of this event decays as $\exp (\Theta(n^2))$. However the question of existence of the rate function i.e., whether the logprobability normalized by $n^2$ 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 $n.$
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.01255
 arXiv:
 arXiv:1712.01255
 Bibcode:
 2017arXiv171201255B
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 40 pages, 12 figures