Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$ between the lower half and the upper half of the box, and the maximal flow $\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\tau$ and $\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\tau /s$ and $\phi /s$ are abnormally small. For $\tau$, the box can have any orientation, whereas for $\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\tau /s$ and $\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables.