English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edgeweights, we consider the upper tail large deviation event; i.e., the raresituation where the first passage time between two points at distance $n$, ismacroscopically larger than typical. It was shown by Kesten (1986) that theprobability of this event decays as $\exp (-\Theta(n^2))$. However the questionof existence of the rate function i.e., whether the log-probability normalizedby $n^2$ tends to a limit, had remained open. We show that under someadditional mild regularity assumption on the passage time distribution, therate function for upper tail large deviation indeed exists. Our proof can begeneralized to work in higher dimensions and for the corresponding problem inlast passage percolation as well. The key intuition behind the proof is that alimiting metric structure which is atypical causes the upper tail largedeviation event. The formal argument then relies on an approximate version ofthe above which allows us to dilate the large deviation environment to comparethe upper tail probabilities for various values of $n.$

Upper Tail Large Deviations in First Passage Percolation