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Lower bounds on non-random fluctuations in planar first passage percolation

Malte Hassler

TL;DR

The paper proves a lower bound of order $\log(|x|)^{1/2-\kappa}$ for the non-random fluctuations in planar first passage percolation under broad edge-weight distributions, advancing beyond the prior $\log\log(n)$ bound. The approach fuses the BKS midpoint framework with Mermin-Wagner type estimates to generate controllable, low-cost fluctuations near the origin that influence the two-point passage time. This establishes divergence of non-random fluctuations in all planar directions, under mild regularity assumptions (AC or near-AC weights) and suggests directions for potential strengthening under local deviation hypotheses. The results illuminate the interplay between geodesic structure, midpoint phenomena, and fluctuation lower bounds in two-dimensional FPP, with implications for KPZ-type universality and limit-shape considerations.

Abstract

The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order $\log(n)^α$ for any $α<1/2$ under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the ${\log(\log(n))}$ bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions. Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.

Lower bounds on non-random fluctuations in planar first passage percolation

TL;DR

The paper proves a lower bound of order for the non-random fluctuations in planar first passage percolation under broad edge-weight distributions, advancing beyond the prior bound. The approach fuses the BKS midpoint framework with Mermin-Wagner type estimates to generate controllable, low-cost fluctuations near the origin that influence the two-point passage time. This establishes divergence of non-random fluctuations in all planar directions, under mild regularity assumptions (AC or near-AC weights) and suggests directions for potential strengthening under local deviation hypotheses. The results illuminate the interplay between geodesic structure, midpoint phenomena, and fluctuation lower bounds in two-dimensional FPP, with implications for KPZ-type universality and limit-shape considerations.

Abstract

The fluctuations of the passage time in first passage percolation are of great interest. We show that the non-random fluctuations in planar FPP are at least of order for any under some conditions that are known to be met for a large class of absolutely continuous edge weight distributions. This improves the bound proven by Nakajima and is the first result showing divergence of the fluctuations for arbitrary directions. Our proof is an application of recent work by Dembin, Elboim and Peled on the BKS midpoint problem and the development of Mermin-Wagner type estimates.

Paper Structure

This paper contains 7 sections, 3 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Idea of the proof
  4. Outlook
  5. Notation
  6. Some auxiliary results
  7. proof of main result

Key Result

Lemma 1

Let $\nu$ be an absolutely-continuous probability measure on $\mathbb R$. There exist such that the following holds:

Figures (1)

  • Figure 1: The geodesics from $(-n,0)$ to $(n,0)$ are the same for both environments $\omega$ and $\tilde{\omega}$ because they do not intersect $\Lambda(n^{1/33})$. But for geodesics going through the origin, the passage time for $\omega$ is larger. Since $\tilde{\omega} \in A_3^K$, the geodesic $\gamma(n,0)(\omega)$ must go through edges $e$ in each of the rectangular annuli of width $n^{1/66}$ where the edge weight is at least $\tau_e$ larger compared to $\tilde{\omega}$.

Theorems & Definitions (7)

  • Lemma 1: BKS, Lemma 2.12
  • Lemma 2: midpoint problem
  • proof
  • Lemma 3: supercriticality implies positive ratio
  • proof
  • proof
  • proof