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The directed landscape is a black noise

Zoe Himwich, Shalin Parekh

TL;DR

The paper proves that the directed landscape is a black noise in the sense of Tsirelson and Vershik, implying that the driving white noise becomes asymptotically independent of the height profile under KPZ scaling. It reduces the problem to a variance bound for the Airy sheet, establishes a strong $\alpha$-mixing property with rate $e^{-d k^3}$, and develops a suite of Airy-sheet, Airy-process, and three-dimensional Bessel process estimates to carry the argument through. The authors then show that the directed landscape cannot be realized as an SPDE driven by space-time Gaussian white noise, and they prove a decoupling result for KPZ height profiles from the underlying environment. These findings illuminate noise-sensitivity phenomena in KPZ universality and distinguish the strong-disorder regime from intermediate-disorder and weak KPZ scaling regimes. The results have implications for the universality of the KPZ fixed point and the behavior of microscopic noise in scaling limits.

Abstract

We show that the directed landscape is a black noise in the sense of Tsirelson and Vershik. As a corollary, we show that for any microscopic system in which the height profile converges in law to the directed landscape, the driving noise is asymptotically independent of the height profile. This decoupling result provides one answer to the question of what happens to the driving noise in the limit under the KPZ scaling, and illustrates a type of noise sensitivity for systems in the KPZ universality class. Such decoupling and sensitivity phenomena are not present in the intermediate-disorder or weak-asymmetry regime, thus illustrating a contrast from the weak KPZ scaling regime. Along the way, we prove a strong mixing property for the directed landscape on a bounded time interval under spatial shifts, with a mixing rate $α(N)\leq Ce^{-dN^3}$ for some $C,d>0$.

The directed landscape is a black noise

TL;DR

The paper proves that the directed landscape is a black noise in the sense of Tsirelson and Vershik, implying that the driving white noise becomes asymptotically independent of the height profile under KPZ scaling. It reduces the problem to a variance bound for the Airy sheet, establishes a strong -mixing property with rate , and develops a suite of Airy-sheet, Airy-process, and three-dimensional Bessel process estimates to carry the argument through. The authors then show that the directed landscape cannot be realized as an SPDE driven by space-time Gaussian white noise, and they prove a decoupling result for KPZ height profiles from the underlying environment. These findings illuminate noise-sensitivity phenomena in KPZ universality and distinguish the strong-disorder regime from intermediate-disorder and weak KPZ scaling regimes. The results have implications for the universality of the KPZ fixed point and the behavior of microscopic noise in scaling limits.

Abstract

We show that the directed landscape is a black noise in the sense of Tsirelson and Vershik. As a corollary, we show that for any microscopic system in which the height profile converges in law to the directed landscape, the driving noise is asymptotically independent of the height profile. This decoupling result provides one answer to the question of what happens to the driving noise in the limit under the KPZ scaling, and illustrates a type of noise sensitivity for systems in the KPZ universality class. Such decoupling and sensitivity phenomena are not present in the intermediate-disorder or weak-asymmetry regime, thus illustrating a contrast from the weak KPZ scaling regime. Along the way, we prove a strong mixing property for the directed landscape on a bounded time interval under spatial shifts, with a mixing rate for some .
Paper Structure (14 sections, 35 theorems, 139 equations, 1 figure)

This paper contains 14 sections, 35 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preface
  3. Background and Results
  4. Black Noise
  5. The KPZ Universality Class and the directed landscape
  6. Notation
  7. Structure of the Paper
  8. Acknowledgements
  9. Reducing the black noise problem to an estimate on the Airy sheet
  10. A strong mixing property for the landscape under spatial shifts
  11. Useful estimates on Airy sheets and argmaxes
  12. Estimates on three-dimensional Bessel processes
  13. Proof of the main result: Theorem \ref{['t:1']}
  14. Independence from Gaussian white noise

Key Result

Lemma 1.9

The directed landscape defines a noise $(\Omega^{\mathrm{DL}},\left(\mathcal{F}^{\mathrm{DL}}_{s,t}\right)_{s<t},\mathbb{P},\left(\theta_{h}^{\mathrm{DL}}\right)_{h})$ in the sense of d:noise, where $(\theta_{h}^{\mathrm{DL}} \mathcal{L})_{s,t}(x,y):= \mathcal{L}_{s+h,t+h}(x,y)$.

Figures (1)

  • Figure 1: A visual depiction of the "geodesic separation" argument in the proof of \ref{['mixing']}. We view the horizontal axis as a spatial axis, and likewise view the vertical axis as a temporal one. If all of the prelimiting geodesic paths from $b_r$ to $b_r$ and from $c_r$ to $c_r$ do not intersect the dotted lines or the endpoints of the intervals, then the non-crossing property of the geodesics in the prelimiting model implies that all last passage values between two space-time points inside the left shaded region will agree with those determined by $\omega^1$, while all last passage values between two space-time points inside the right shaded region will agree with those determined by $\omega^2$.

Theorems & Definitions (78)

  • Definition 1.3: Definition 3d(1), Tsir
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Lemma 1.9
  • proof
  • Theorem 1.10
  • Theorem 1.11
  • ...and 68 more