Estimates for the empirical distribution along a geodesic in first-passage percolation

In first-passage percolation, we assign i.i.d.~nonnegative weights $(t_e)$ to the nearest-neighbor edges of $\mathbb{Z}^d$ and study the induced pseudometric $T = T(x,y)$. In this paper, we focus on geodesics, or optimal paths for $T$, and estimate the empirical distribution of weights along them. We prove an upper bound for the expected number of edges with weight $\geq M$ in the union of all geodesics from $0$ to $x$ of the form $q(M) \mathbb{P}(t_e \geq M)|x|$, where $q(M) \leq e^{-cM}$. This shows that the tail of the expected empirical distribution along a geodesic is lighter than that of the original weight distribution by an exponential factor. We also give a lower bound for the expected minimal number of edges with weight $\geq M$ in any geodesic from $0$ to $x$ in terms of $\mathbb{P}(t_e \geq M)$ and $\mathbb{P}(t_e \in [M,2M])$. For example, these two imply that if $t_e$ has a power law tail of the form $\mathbb{P}(t_e \geq M) \sim M^{-\alpha}$, then the tail of the expected empirical distribution asymptotically lies between $e^{-CM \log M}$ and $e^{-cM}$. We also provide estimates for the expected number of edges in a geodesic with weight in a set $A$ for (a) arbitrary $A$, (b) $A$ an interval separated from the infimum of the support of $t_e$ and (c) $A=[0,a]$ for some $a \geq 0$.