Moderate deviations in first-passage percolation for bounded weights

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.