Upper Tail Large Deviations in First Passage Percolation

Abstract

For first passage percolation on ℤ2 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()−normalΘ()n2. However, the question of existence of the rate function, i.e., whether the log‐probability normalized by n2 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.

Large deviation principle for the streams and the maximal flow in first passage percolation

“…We refer to [16] and [1] for reviews on the subject. The result in [2] answers an old open question that was first formulated by Kesten in [16]. The correct order of large deviations was already known (see [16]).…”

“…As the capacities are random, the set of admissible streams S n (Γ 1 , Γ 2 , Ω) is also random. We denote by S M n (Γ 1 , Γ 2 , Ω) the set of streams f n : E d n → R d such that • for each edge e = x, y such that (x, y) / ∈ Ω 2 n \ (Γ 1 n ∪ Γ 2 n ) 2 we have f n (e) = 0 • for each e ∈ E d n we have f n (e) 2 ≤ M • the stream satisfies the node law for any vertex x ∈ Z d n \ (Γ 1 n ∪ Γ 2 n ). Note that the set S M n (Γ 1 , Γ 2 , Ω) is a deterministic set.…”

“…Let E ⊂ E d be a set of edges. We say that E cuts Γ 1 from Γ 2 in Ω (or is a cutset, for short) if there is no path from Γ 1 1 to Γ 2 1 in (Ω 1 , E d \ E). More precisely, let γ be a path from Γ 1 1 to Γ 2 1 in Ω 1 , we can write γ as a finite sequence (v 0 , e 1 , v 1 , .…”

“…Let us now define the distance d. Let k ∈ N. Let λ ∈ [1,2]. We denote by ∆ k λ the set of dyadic cubes at scale k with scaling parameter λ, that is,…”

“…Our paper finds its inspiration in the philosophy of [2]: we use large deviations events at a small scale to build large deviations events at a higher scale. We here manage to formalize the idea of a limiting environment.…”