Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From stability to chaos in last-passage percolation

Daniel Ahlberg, Maria Deijfen, Matteo Sfragara

Abstract

We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint.

From stability to chaos in last-passage percolation

Abstract

We study the transition from stability to chaos in a dynamic last passage percolation model on with random weights at the vertices. Given an initial weight configuration at time , we perturb the model over time in such a way that the weight configuration at time is obtained by resampling each weight independently with probability . On the cube , we study geodesics, that is, weight-maximizing up-right paths from to , and their passage time . Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at . Indeed, as grows large, for small values of , the passage times at time and time are highly correlated, while for large values of , the geodesics become almost disjoint.
Paper Structure (10 sections, 4 theorems, 46 equations)

This paper contains 10 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Model and results
  4. Discussion
  5. The covariance formula
  6. Bounding the influence
  7. From stability to chaos
  8. Proof of Theorem \ref{['thm:stabilityandchaos']}, part (i)
  9. Proof of Theorem \ref{['thm:stabilityandchaos']}, part (ii)
  10. Appendix: Distributions satisfying \ref{['assumption2']}

Key Result

Theorem 1.1

Consider last-passage percolation on $\mathbb{Z}^d$, for $d \geq 2$, with a weight distribution satisfying Fass. There exists a constant $C<\infty$ such that for all $n\ge1$ and $0<\alpha<\frac{n}{\mathrm{Var}(T)}$ the following two statements hold.

Theorems & Definitions (9)

  • Theorem 1.1: From stability to chaos
  • Conjecture 1.2
  • Conjecture 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2: Upper bound
  • proof
  • Lemma 3.3: Lower bound
  • proof