Moderate deviations in first-passage percolation for bounded weights

TL;DR

This paper studies the probabilities of moderate deviations in first-passage percolation with bounded weights, establishing new asymptotic formulas and concentration results that connect fluctuations with large deviation decay rates.

Contribution

It provides the first rigorous estimates for moderate deviations in FPP with bounded weights, linking these to large deviation rate functions and improving concentration bounds.

Findings

Derived asymptotic formulas for moderate deviations probabilities

Established estimates for large deviation tail behaviors

Enhanced concentration inequalities via multi-scale analysis

Abstract

We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $\mu(\mathbf{u})$ the time constant in direction $\mathbf{u}$. In this paper, we establish that, if one assumes that the sublinear error term $T(\mathbf{0}, N\mathbf{u}) - N\mu(\mathbf{u})$ is of order $N^\chi$, then under some unverified (but widely believed) assumptions, for $\chi < a < 1$, \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > N\mu(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-\chi}(a-\chi)}\Big)},\end{align*} \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < N\mu(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-\chi}(a-\chi)}\Big)}, \end{align*} with accompanying estimates in the borderline case $a=1$. Moreover, the exponents $\frac{d}{1-\chi}$ and $\frac{1}{1-\chi}$ also appear in the asymptotic behavior near $0$ of the rate functions for upper and lower tail large deviations. Notably, some of our estimates are established rigorously without relying on any unverified assumptions. Our main results highlight the interplay between fluctuations and the decay rates of large deviations, and bridge the gap between these two regimes. A key ingredient of our proof is an improved concentration via multi-scale analysis for several moderate deviation estimates, a phenomenon that has previously appeared in the contexts of two-dimensional last-passage percolation and two-dimensional rotationally invariant FPP.