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Strict inequality between the time constants of first-passage percolation and directed first-passage percolation

Antonin Jacquet

TL;DR

The authors establish a strict separation between the undirected and directed time constants in first-passage percolation on $\mathbb{Z}^d$ by proving an exponential bound that compares $t(0,x)$ and $\vec{t}(0,x)$ as $\|x\|_1$ grows. Central to the approach is a lower bound on the Euclidean length of geodesics, obtained via a patterns-based argument that forces geodesics to cross a linear number of certain configurations. Building on this, they derive an exponential bound for the probability that $t(0,x)$ falls below $\vec{t}(0,x)$ by a fixed margin, and then deduce $\mu(x) < \vec{\mu}(x)$ for all $x\ge0$, $x\neq0$. The results hold under the notion of a useful passage-time distribution and suitable percolation conditions, and they relate to prior work by KRAS on the structure of geodesics and time constants in FPP.

Abstract

In the models of first-passage percolation and directed first-passage percolation on $\mathbb{Z}^d$, we consider a family of i.i.d. random variables indexed by the set of edges of the graph, called passage times. For every vertex $x \in \mathbb{Z}^d$ with nonnegative coordinates, we denote by $t(0,x)$ the shortest passage time to go from $0$ to $x$ and by $\vec t(0,x)$ the shortest passage time to go from $0$ to $x$ following a directed path. Under some assumptions, it is known that for every $x \in \mathbb{R}^d$ with nonnegative coordinates, $t(0,\lfloor nx \rfloor)/n$ converges to a constant $μ(x)$ and that $\vec t(0,\lfloor nx \rfloor)/n$ converges to a constant $\vecμ(x)$. With these definitions, we immediately get that $μ(x) \le \vecμ(x)$. In this paper, we get the strict inequality $μ(x) < \vecμ(x)$ as a consequence of a new exponential bound for the comparison of $t(0,x)$ and $\vec{t}(0,x)$ when $\|x\|$ goes to $\infty$. This exponential bound is itself based on a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time).

Strict inequality between the time constants of first-passage percolation and directed first-passage percolation

TL;DR

The authors establish a strict separation between the undirected and directed time constants in first-passage percolation on by proving an exponential bound that compares and as grows. Central to the approach is a lower bound on the Euclidean length of geodesics, obtained via a patterns-based argument that forces geodesics to cross a linear number of certain configurations. Building on this, they derive an exponential bound for the probability that falls below by a fixed margin, and then deduce for all , . The results hold under the notion of a useful passage-time distribution and suitable percolation conditions, and they relate to prior work by KRAS on the structure of geodesics and time constants in FPP.

Abstract

In the models of first-passage percolation and directed first-passage percolation on , we consider a family of i.i.d. random variables indexed by the set of edges of the graph, called passage times. For every vertex with nonnegative coordinates, we denote by the shortest passage time to go from to and by the shortest passage time to go from to following a directed path. Under some assumptions, it is known that for every with nonnegative coordinates, converges to a constant and that converges to a constant . With these definitions, we immediately get that . In this paper, we get the strict inequality as a consequence of a new exponential bound for the comparison of and when goes to . This exponential bound is itself based on a lower bound on the number of edges of geodesics in first-passage percolation (where geodesics are paths with minimal passage time).
Paper Structure (7 sections, 4 theorems, 23 equations)

This paper contains 7 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction and results
  2. The model of first-passage percolation.
  3. Euclidean length of geodesics in first-passage percolation
  4. Directed first-passage percolation
  5. Ideas of proofs
  6. Proofs of Theorem \ref{['thm: Théorème principal bis 2.']} and Theorem \ref{['thm: Théorème principal.']}
  7. Proof of Theorem \ref{['thm: Théorème principal 2.']}

Key Result

Theorem 1.1

Assume that the support of $\mathcal{L}$ is included in $[0,\infty]$, that $\mathcal{L}$ is useful (i.e. that $\mathcal{L}$ satisfies eq: loi useful.) and that $\mathcal{L}([0,\infty)) > p_c$. There exist deterministic constants $\alpha_1>0$, $\alpha_2>0$ and $\delta > 0$ such that for all $x \in \m

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • proof : Proof of Theorem \ref{['thm: Théorème principal bis 2.']} using Theorem \ref{['thm: Théorème principal 2.']}
  • proof : Proof of Theorem \ref{['thm: Théorème principal.']}
  • proof : Proof of Theorem \ref{['thm: Théorème principal 2.']}