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Noise sensitivity in last-passage percolation

Daniel Ahlberg, Malo Hillairet, Ekaterina Toropova

TL;DR

This work establishes noise sensitivity for planar last-passage percolation in the KPZ class by proving a real-valued generalization of the BKS influence theorem and bounding the probability that a vertex lies on a geodesic. The authors encode geometric weights via a Bernoulli bit-encoding and prove that the travel time $T_n$ is noise sensitive under bit-resampling, with a quantitative covariance bound linking noise strength to vertex-inclusion probabilities. A stationary LPP framework with boundary weights and Busemann functions is developed to bound geodesic visitation both far from and near the diagonal, enabling a precise control of the total influence and the transversal fluctuation exponent. The paper further compares bit-resampling and site-resampling noises, discusses hypercontractivity limitations, and outlines open problems, including the conjectured KPZ-threshold scaling $t\asymp n^{-1/3}$ and extensions to Poisson or continuum models, highlighting the broader significance for understanding chaos in spatial growth models.

Abstract

The study of noise sensitivity of Boolean functions was initiated in a seminal paper of Benjamini, Kalai and Schramm, published in 1999. While this study has revealed fascinating phenomena in the context of Bernoulli percolation, few results have been obtained regarding other random spatial processes. In this paper we prove the first instance of noise sensitivity for a spatial growth process associated to the Kardar-Parisi-Zhang class of universality. More specifically, we show that travel times in geometric last-passage percolation are noise sensitive with respect to a perturbation acting on a Bernoulli encoding of the geometric weights. Our method of proof includes a generalisation of the celebrated Benjamini-Kalai-Schramm noise sensitivity/influence theorem, and precise bounds on the probability of a given vertex being on a geodesic, which we believe to be of independent interest.

Noise sensitivity in last-passage percolation

TL;DR

This work establishes noise sensitivity for planar last-passage percolation in the KPZ class by proving a real-valued generalization of the BKS influence theorem and bounding the probability that a vertex lies on a geodesic. The authors encode geometric weights via a Bernoulli bit-encoding and prove that the travel time is noise sensitive under bit-resampling, with a quantitative covariance bound linking noise strength to vertex-inclusion probabilities. A stationary LPP framework with boundary weights and Busemann functions is developed to bound geodesic visitation both far from and near the diagonal, enabling a precise control of the total influence and the transversal fluctuation exponent. The paper further compares bit-resampling and site-resampling noises, discusses hypercontractivity limitations, and outlines open problems, including the conjectured KPZ-threshold scaling and extensions to Poisson or continuum models, highlighting the broader significance for understanding chaos in spatial growth models.

Abstract

The study of noise sensitivity of Boolean functions was initiated in a seminal paper of Benjamini, Kalai and Schramm, published in 1999. While this study has revealed fascinating phenomena in the context of Bernoulli percolation, few results have been obtained regarding other random spatial processes. In this paper we prove the first instance of noise sensitivity for a spatial growth process associated to the Kardar-Parisi-Zhang class of universality. More specifically, we show that travel times in geometric last-passage percolation are noise sensitive with respect to a perturbation acting on a Bernoulli encoding of the geometric weights. Our method of proof includes a generalisation of the celebrated Benjamini-Kalai-Schramm noise sensitivity/influence theorem, and precise bounds on the probability of a given vertex being on a geodesic, which we believe to be of independent interest.
Paper Structure (31 sections, 27 theorems, 224 equations, 4 figures)

This paper contains 31 sections, 27 theorems, 224 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Noise sensitivity and influences
  4. Bounding the influences
  5. Discussion of the notion of noise
  6. From stability to chaos in last-passage percolation
  7. Organization of the article
  8. A generalization of the BKS theorem
  9. Noise sensitivity in a semigroup framework
  10. Operator properties
  11. Hypercontractivity
  12. Proof of the generalized BKS theorem
  13. Non-hypercontractive semi-groups
  14. Stationary last-passage percolation
  15. Some notation
  16. ...and 16 more sections

Key Result

Theorem 1.1

For $n\ge1$, let $T_n = T_n(\omega)$ denote the last-passage time from $(0, 0)$ to $(n, n)$ with respect to the weight configuration $\omega$. Planar last-passage percolation with geometric weights is noise sensitive in the sense that, for every $p\in(0,1)$ and $t>0$,

Figures (4)

  • Figure 1: The directed path $\gamma$ is above $\gamma'$.
  • Figure 2: Illustration of Theorem \ref{['backgeodlargedev']}: with probability at least $1 - e^{-c s^3}$, the intersections of $\pi(\lambda; - \infty, 0)$ with $H_{- \lfloor r \mathbf{u}_{\lambda} \rfloor}$ are at distance less than $s r^{2/3}$ from $-r \mathbf{u}_{\lambda}$.
  • Figure 3: Illustration of $v'$ and $v"$ and the shaded right quadrant of $R_{-n \mathbf{e}_+, 0}$ (left). Illustration of the intervals around $v'$ and $v"$ which $\pi(\lambda; - \infty, 0)$ goes through with high probability (right).
  • Figure 4: Schematic representation of the directed geodesics given by $\lambda^+$ and $\lambda^-$

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Conjecture 1.7
  • Lemma 2.1: Commutativity
  • proof
  • Lemma 2.2: Heat equation
  • ...and 45 more