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

Abstract

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 $\ge M$ in the union of all geodesics from $0$ to $x$ of the form $q(M)\mathbb{P}(t_e\ge M)|x|$, where $q(M)\le e^{-cM}$. 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 $\ge M$ in any geodesic from $0$ to $x$ in terms of $\mathbb{P}(t_e\ge M)$ 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 $\mathbb{P}(t_e\ge M)\sim M^{-\hat{I}\pm }$, 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 $a\ge 0$.

3 figures, 38 pages. Minor updates compared to previous version following referee comments