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

TL;DR

This paper advances the understanding of empirical edge-weight distributions along geodesics in first-passage percolation by deriving sharp tails and linear-in-distance bounds. It combines percolation techniques, stability arguments (via a variable $D_e$), edge-modification resampling, and lattice-animal bounds to show that the tail of the empirical distribution along geodesics is exponentially lighter than the original tail, with precise bounds that adapt to the tail of $t_e$. It further extends previous work by providing an exponent-1/d improvement for general edge-weight sets and linear bounds when sets are separated from the infimum of the weight distribution, along with a BK-type lower bound for $N_{\min}$ and a FKG-based lower bound for small-weight sets. The results illuminate how the geometric structure of geodesics constrains the spread of edge-weights they encounter, with implications for heavy- and light-tailed distributions and for a broad class of weight sets $A$. These findings contribute both to the theoretical landscape of FPP and to potential applications in understanding transport properties in random media.

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 $\mathbb{P}(t_e \geq M) \sim M^{-α}$, 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 $a \geq 0$.

Figures (3)

• Figure 1: Illustration of the event that the edge $e$ is good. The paths $\pi_1$ and $\pi_2$ begin at $e$ and end at vertices in $\partial B(e,k)$. The path $\pi$ connects a vertex $x_1$ on $\pi_1$ to a vertex $x_2$ on $\pi_2$, and has at most $\Cr{c: good_edge} k$ many edges. All edges on $\pi$ have weight at most $M_0$ and $e$ is not an edge of $\pi$.
• Figure 2: Illustration of path and edge set definitions. In top-left, $u$ and $v$, the first and last intersections of $\gamma_{x,y}$ with $R(n)$, are far apart ($\|u-v\|_1 \geq n/2$) and so the path $\overline{\pi}_{u,v}$ from $\overline{u}$ to $\overline{v}$ is chosen to be oriented. In top-right, $u$ and $v$ are close ($\|u-v\|_1 < n/2$) and so $\overline{\pi}_{u,v}$ is a concatenation of two oriented paths. In both cases, $u$ is the corner vertex $(0,3n)$ and is not adjacent to $R(n) \setminus \partial_iR(n)$, so it cannot be the endpoint of a rung, and we choose $u^\ast = (1,3n)$. In both cases, we choose $v = v^\ast$ and so $\pi_{u,v}^{(2)}$ is empty. The rung $\pi_{u,v}$ (not labeled) moves from $u^\ast$ to $\overline{u}$, follows $\overline{\pi}_{u,v}$, then moves from $\overline{v}$ to $v^\ast$. The path $\sigma_{u,v}$ (not labeled) follows $\pi_{u,v}^{(1)}$, then $\pi_{u,v}$, and then $\pi_{u,v}^{(2)}$. The bottom two subfigures illustrate the sets $E_i(u,v)$ for $i=1, \dots, 4$ and correspond to the cases shown in the subfigures directly above them. Note that $\overline{\pi}_{u,v}$ has length $<n$ in the two right figures, although due to space constraints, the path appears longer than the width $n$ of the box.
• Figure 3: Illustration of the argument for Lem. \ref{['lem: another_good_lemma']}. The paths $\gamma_1$ and $\gamma_2$ are optimal paths for $T(w_1,\partial B(\Cr{c: good_constant}n))$ and $T(w_2,\partial B(\Cr{c: good_constant}n))$. The path $\gamma_3$ is contained in the infinite component $I$ and connects points $c$ on $\gamma_1$ and $d$ on $\gamma_2$. Because the union of $\gamma_3$ with the initial segments of $\gamma_1$ and $\gamma_2$ comprises a path from $w_1$ to $w_2$ with smaller passage time than the sum of $T(\gamma_1)$ and $T(\gamma_2)$, no geodesic from $w_1$ to $w_2$ may exit $B(\Cr{c: good_constant}n)$.

Theorems & Definitions (32)

• Theorem 1.1
• Remark 1
• Remark 2
• Theorem 1.2
• Remark 3
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Proposition 2.1
• Lemma 2.2