Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of the limit shape for 1+1-dimensional FPP

Malte Hassler

TL;DR

This work introduces and analyzes a simplified 1+1-dimensional first passage percolation model with deterministic vertical weights and i.i.d. horizontal weights drawn from $G$. It proves a sharp criterion for when the limit shape has a vertical flat edge: the edge at $(0,1)$ occurs if and only if $G$ has an atom at the infimum $t_0$ of its support, i.e., $G(\\{t_0\\})>0$, with $t_0=\\inf\{x: G([0,x])>0\}$. The authors develop a pair of complementary derivative bounds for the time constant $\\Lambda$, expressing these derivatives in terms of the local geometry of geodesics via pioneer points and turn counts, and they recast the problem using an equivalent polymer-like model and a random-shearing mechanism. The paper also establishes regularity of the passage time, provides upper/lower bounds on the limit shape, and proves auxiliary results on large deviations and concentration, which collectively deepen understanding of geodesic structure and limit-shape behavior in this discrete, non-integrable setting. Overall, the results connect limit-shape regularity and flat-edge phenomena to the microscopic atom structure of the horizontal-edge distribution, with implications for geodesic configurations and potential extensions to more general FPP-like models.

Abstract

We introduce a simplified model of planar first passage percolation where weights along vertical edges are deterministic. We show that the limit shape has a flat edge in the vertical direction if and only if the random distribution of the horizontal edges has an atom at the infimum of its support. Furthermore, we present bounds on the upper and lower derivative of the time constant.

Classification of the limit shape for 1+1-dimensional FPP

TL;DR

This work introduces and analyzes a simplified 1+1-dimensional first passage percolation model with deterministic vertical weights and i.i.d. horizontal weights drawn from . It proves a sharp criterion for when the limit shape has a vertical flat edge: the edge at occurs if and only if has an atom at the infimum of its support, i.e., , with . The authors develop a pair of complementary derivative bounds for the time constant , expressing these derivatives in terms of the local geometry of geodesics via pioneer points and turn counts, and they recast the problem using an equivalent polymer-like model and a random-shearing mechanism. The paper also establishes regularity of the passage time, provides upper/lower bounds on the limit shape, and proves auxiliary results on large deviations and concentration, which collectively deepen understanding of geodesic structure and limit-shape behavior in this discrete, non-integrable setting. Overall, the results connect limit-shape regularity and flat-edge phenomena to the microscopic atom structure of the horizontal-edge distribution, with implications for geodesic configurations and potential extensions to more general FPP-like models.

Abstract

We introduce a simplified model of planar first passage percolation where weights along vertical edges are deterministic. We show that the limit shape has a flat edge in the vertical direction if and only if the random distribution of the horizontal edges has an atom at the infimum of its support. Furthermore, we present bounds on the upper and lower derivative of the time constant.

Paper Structure

This paper contains 24 sections, 35 theorems, 115 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model and main result
  3. Motivation and organization of the paper
  4. Connecting the derivative of the time constant with the local geometry of geodesics
  5. Geodesics
  6. The derivative of the time constant
  7. An equivalent model
  8. Random shearing
  9. The upper derivative
  10. The lower derivative
  11. Classification of the limit shape
  12. Regularity of the passage time
  13. Bounds on the limit shape
  14. The local behavior of semi-directed paths
  15. Staying above the asymptote
  16. ...and 9 more sections

Key Result

Theorem 1

Let $G$ be any probability measure on $[0, \infty)$. Then there exists a convex, deterministic function $\Lambda: [0,\infty) \to [0, \infty)$ depending on $G$ such that almost surely and in $L^1$.

Figures (2)

  • Figure 1: A semi-directed path $p$ from the origin to $(n, {\lceil v n \rceil})$ with $n=7$ and $v=0.7$. Pioneer points are shown in green and $f_p=(0,2,2,1,-1,-1,4,6,5)$.
  • Figure 2: A semi-directed path starting from the origin together with the associated macroscopic vertices from $\mathcal{A}_K$ (blue) and $\mathcal{A}_K'$ (red). Shaded regions indicate a trapezoidal crossing.

Theorems & Definitions (60)

  • Theorem 1
  • Lemma 1
  • Definition 1
  • Theorem 2
  • Theorem Main Theorem
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 50 more