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

TL;DR

This paper analyzes the distribution of edge weights along geodesics in first-passage percolation, providing bounds showing the empirical distribution's tail is lighter than the original, with implications for power-law tails.

Contribution

It introduces bounds on the empirical distribution of weights along geodesics, revealing exponential decay in the tail and offering new insights into the structure of optimal paths.

Findings

Expected number of edges with weight ≥ M is bounded by q(M) |x|, with q(M) ≤ e^{-cM}

Tail of empirical distribution along geodesics decays faster than original distribution, e.g., e^{-CM log M} for power-law tails

Provides estimates for edges with weights in specific sets, including intervals and bounded sets

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 $\geq M$ 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 $a \geq 0$.