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Brochette first-passage percolation

Maxime Marivain

TL;DR

This work introduces the Brochette first-passage percolation model, in which edge times are line-dependent but line-irreducible across lines, and establishes a time constant $a\|x\|_1$ under a finite-min moment condition. It proves a shape theorem with the $L^1$ diamond as the limit shape when $a>0$, and provides a detailed analysis for the critical case $a=0$ in dimension two, where the near-zero tail of the distribution drives a spectrum of asymptotic behaviors. The article also extends the framework to infinite passage times by regularizing on finite clusters and proving convergence in probability for the effective distance $T^{*}$, with an auxiliary appendix detailing the asymptotics of the minimum of i.i.d. samples. The results combine subadditive ergodic techniques, geometric skeleton constructions, and concentration-type arguments to handle long-range line-based dependence unique to the Brochette construction, offering a comprehensive view of how line-wise dependence alters growth, shaping, and phase-like behaviors in FPP.

Abstract

We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the $L^1$ diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.

Brochette first-passage percolation

TL;DR

This work introduces the Brochette first-passage percolation model, in which edge times are line-dependent but line-irreducible across lines, and establishes a time constant under a finite-min moment condition. It proves a shape theorem with the diamond as the limit shape when , and provides a detailed analysis for the critical case in dimension two, where the near-zero tail of the distribution drives a spectrum of asymptotic behaviors. The article also extends the framework to infinite passage times by regularizing on finite clusters and proving convergence in probability for the effective distance , with an auxiliary appendix detailing the asymptotics of the minimum of i.i.d. samples. The results combine subadditive ergodic techniques, geometric skeleton constructions, and concentration-type arguments to handle long-range line-based dependence unique to the Brochette construction, offering a comprehensive view of how line-wise dependence alters growth, shaping, and phase-like behaviors in FPP.

Abstract

We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.
Paper Structure (10 sections, 18 theorems, 90 equations, 5 figures)

This paper contains 10 sections, 18 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. First-passage percolation in the independent setting
  3. Brochette first-passage percolation
  4. The results
  5. Point to point almost sure convergence
  6. The case $a=0$
  7. Shape Theorem
  8. Construction of paths $\Gamma_{1},\Gamma_2,\Gamma_3$.
  9. Brochette First-Passage Percolation with infinite passage times
  10. Appendix

Key Result

Theorem 1.1

Assume that $(\tau_{e})_{e\in\mathbb{E}^{d}}$ are i.i.d. random variables such that $\mathbb{E}[\min(\tau_{1},...,\tau_{2d})]<+\infty$, then for all $x\in\mathbb{Z}^d$ there exists a constant $\mu(x)\geq0$ (called the time constant) such that:

Figures (5)

  • Figure 1: The random balls $B_t$ for $t=500$ for the distribution $1+Bernoulli(0.95)$ for the classical model (on the left) and the Brochette model (on the right).
  • Figure 2: The path going from $0$ to $ne_1$ following the construction.
  • Figure 3: Paths $\Gamma_1,\Gamma_2,\Gamma_3$.
  • Figure 4: Envelopes which are dependent of the red one.
  • Figure 5:

Theorems & Definitions (40)

  • Theorem 1.1: kingmanpap, see Theorem 2.1 Survey
  • Theorem 1.2: Cox
  • Definition 1.3
  • Definition 1.4: Brochette first-passage percolation
  • Theorem 1.5: Point to point convergence theorem
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 30 more