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The KPZ fixed point and Brownian motion share the same null sets

Pantelis Tassopoulos, Sourav Sarkar

Abstract

We show that the increments of the KPZ fixed point started from arbitrary initial data are \emph{mutually} absolutely continuous with respect to Brownian motion with diffusion parameter $2$ on compacts, extending the one-sided Brownian absolute continuity relation of the KPZ fixed point established in \cite{sarkar2021brownian}. We also show that additive Brownian motion is absolutely continuous with respect to the centred Airy sheet on compacts, but it is not mutually absolutely continuous globally. As applications, we show that with probability strictly between zero and one, there exist record times of the KPZ fixed point away from any reference point, obtain a characterisation for the hitting probabilities of the graph of the KPZ fixed point to be positive in terms of a certain thermal capacity in the sense of \cite{watson1978corrigendum, watson1978thermal} and compute essential suprema of Hausdorff dimensions of these random intersections. Finally, we compute essential suprema of Hausdorff dimensions of images of subsets in the plane under the Airy sheet and give a condition for the positivity of their Lebesgue measure in terms of Bessel-Riesz capacity.

The KPZ fixed point and Brownian motion share the same null sets

Abstract

We show that the increments of the KPZ fixed point started from arbitrary initial data are \emph{mutually} absolutely continuous with respect to Brownian motion with diffusion parameter on compacts, extending the one-sided Brownian absolute continuity relation of the KPZ fixed point established in \cite{sarkar2021brownian}. We also show that additive Brownian motion is absolutely continuous with respect to the centred Airy sheet on compacts, but it is not mutually absolutely continuous globally. As applications, we show that with probability strictly between zero and one, there exist record times of the KPZ fixed point away from any reference point, obtain a characterisation for the hitting probabilities of the graph of the KPZ fixed point to be positive in terms of a certain thermal capacity in the sense of \cite{watson1978corrigendum, watson1978thermal} and compute essential suprema of Hausdorff dimensions of these random intersections. Finally, we compute essential suprema of Hausdorff dimensions of images of subsets in the plane under the Airy sheet and give a condition for the positivity of their Lebesgue measure in terms of Bessel-Riesz capacity.

Paper Structure

This paper contains 18 sections, 29 theorems, 144 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Organization of the paper
  4. Notation
  5. Preliminaries
  6. Last passage percolation
  7. Pitman transform
  8. Brownian bridge properties
  9. Airy line ensemble and the Brownian Gibbs property
  10. The Airy sheet and the directed landscape
  11. The KPZ fixed point
  12. Brownian mutual absolute continuity of the KPZ fixed point
  13. The Airy Sheet and additive Brownian motion
  14. Applications
  15. Record times for the KPZ fixed point
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $t>0$, $-\infty< y_1< y_2<\infty$; then for an arbitrary admissible initial condition $h_0$, the law of $\mathfrak{h}_t(y)-\mathfrak{h}_t(y_1)$ for $y\in [y_1,y_2]$, where $\mathfrak{h}_t$ is the KPZ fixed point at time $t$ started from $h_0$, is mutually absolutely continuous with respect to th

Figures (7)

  • Figure 1: Left: the centred KPZ fixed point started from flat initial data. Right: Brownian motion with diffusion parameter $2$.
  • Figure 2: Flowchart of main steps in the proof of Theorem \ref{['thm: mut abs cont finitary init data']}.
  • Figure 3: Illustration of the coupling between the KPZ fixed points on $[0,2]$ started from compactly supported flat $0$ initial data on $[0, 1]$, that is $0\cdot \delta_{[0, 1]}$ (recall \ref{['eq: max-plus indicator']}) ( green) and the superposition of two narrow wedges at $0, 1$, that is $0\cdot \delta_{\{0, 1\}}$ ( blue).
  • Figure 4: Illustration of the Brownian Gibbs resampling for the top line of the Airy line ensemble appearing in the variational expression for the KPZ fixed point on the compact interval $[0, r]$. In short, it can be expressed (up to mutual absolute continuity) as a concatenation of a Brownian bridge $W$ and Brownian motion $B$ (conditionally independent given the Airy line ensemble) at the point $(\varepsilon, \mathcal{A}_1(\varepsilon-\mathcal{A}_1(0)+G_1)$, conditioned to not hit $\mathcal{A}_2$. The increments of the KPZ fixed point in the interior interval $[\varepsilon, r]$ on the event $\mathcal{A}_1(\cdot)-\mathcal{A}_1(0)+G_1$ avoids $\max_{\ell \ge 2}(G_\ell + \mathcal{A}[(0, \ell)\to (\cdot,2)])$ are simply the increments of the Brownian motion $B$. By standard monotonicity results for Brownian bridges, conditionally on $B|_{[\varepsilon, r]}$, the above non-intersection condition can always be ensured to occur with positive probability.
  • Figure 5: Illustration of the KPZ fixed point at unit time, $\mathfrak{h}_1(\cdot, h_0)$ on the compact interval $[0, y_0]$ in Theorem \ref{['thm: mut abs cont finitary init data']} on the event the top lines of the Pitman transforms in \ref{['eq: pitman melon fin init KPZ']}, $F_1(\cdot) \lor G_1(\cdot)$ does not hit $F_2(\cdot) \lor G_2(\cdot)$. On this event, $\mathfrak{h}_1(\cdot, h_0)$ is exactly equal to the Airy$_2$ process up to a height shift that is $\mathscr{G}$-measurable (recall the notation from Theorem \ref{['thm: mut abs cont finitary init data']}. Moreover, the 'lower barrier' $F_2(\cdot) \lor G_2(\cdot)$ is $\mathscr{G}$-measurable. Using the Brownian Gibbs property, one can represent the top line of the Airy line ensemble (up to mutual absolute continuity) as a 'Brownian bridge sandwich' (conditioned to avoid $\mathcal{A}_2$), that is a Brownian bridge starting from $0$ concatenated to a Brownian motion starting from $a$ which itself is concatenated to another Brownian bridge starting from $b$ and ending at $y_0$.
  • ...and 2 more figures

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 2.1: Absolute continuity
  • Definition 2.2: Path
  • Definition 2.3: Length
  • Definition 2.4: Last passage value
  • Remark
  • Lemma 2.5: Metric composition law
  • Lemma 2.6
  • Lemma 2.7
  • Remark
  • ...and 56 more