Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data

TL;DR

The paper proves a quantitative Brownian regularity for the KPZ fixed point, showing that its spatial increments started from finitary initial data are strongly absolutely continuous with respect to a rate two Brownian motion on compacts. The authors develop a comprehensive framework combining the directed landscape, Airy line ensemble with Brownian Gibbs, and inhomogeneous Brownian LPP to control Radon-Nikodym derivatives and geodesic geometry. They establish tail bounds for semi-infinite geodesic intercepts and uniform coalescence depths for speeds drawn from meagre sets, then combine finite-depth KPZ truncations with localisation to extend the results to all finitary initial data. The work provides a rigorous quantitative link between KPZ fluctuations and Brownian motion, with potential L^p-density refinements and broader implications for KPZ universality.

Abstract

We show that the spatial increments of the KPZ fixed point starting from arbitrary initial data, exhibit strong quantitative comparison against rate two Brownian motion on compacts. The above estimates are uniform for uniformly bounded continuous, compactly supported initial data and countably many narrow wedges with supports contained in a fixed compact set.

Figures (5)

• Figure 1: Flowchart of main steps in the proof of Theorem \ref{['thm: KPZ reg finitary']}.
• Figure 2: Visualisation of a possible path (red) “embedded” on the Airy line ensemble, here $(\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3, \mathcal{A}_4)$ from top to bottom, and $m = 1, k = 4$ (see Section \ref{['subsec: Airy line ensemble']}). Here $\Delta_1 = \mathcal{A}_4(t_1)-\mathcal{A}_4(x)$, $\Delta_2 = \mathcal{A}_3(t_2)-\mathcal{A}_3(t_1)$, $\Delta_3 = \mathcal{A}_2(t_3)-\mathcal{A}_2(t_2)$, $\Delta_4 = \mathcal{A}_1(y)-\mathcal{A}_1(t_3)$ and $\ell = \sum_{i=1}^4 \Delta_i.$
• Figure 3: Brownian melon scaling limit. Above is a realisation of the $WB^4$ melon. 'Zooming in' on the parallelogram at small scales and taking the limit as $n\to\infty$ yields the convergence in law to the (parabolic) Airy line ensemble.
• Figure 4: Figure illustrating the Brownian Gibbs property on the first four lines of the parabolic Airy Line ensemble $\mathcal{A} = \{\mathcal{A}_1>\mathcal{A}_2>\dots\}$ (in black) between two points (indicated by the red vertical dashed lines). The blue curves represent resampled versions of first four lines in the ensemble between the endpoints, conditioning on everything else and avoiding the fifth line.
• Figure 5: Above is displayed the point $(0,L_0)$ at which the last passage path $\pi[x'\to y']$ on the Airy line ensemble $\mathcal{A} = (\mathcal{A}_1, \mathcal{A}_2,\cdots)$ ( purple) meets with the axis $\{x=0\}$, where $y'>1$. Here $L_0 = 3$ and the first four lines of $\mathcal{A}$ are shown. The last passage path $\pi[x'\to y']$ is defined in Definition 3.3 in sarkar2021brownian and the definition \ref{['def: semi-inf geo']}.

Theorems & Definitions (79)

• Definition 1.1
• Definition 1.2
• Definition 1.2
• Theorem 1.3: Quantitative Brownian regularity
• Definition 3.1: Random ensemble
• Definition 3.2: Path
• Remark
• Definition 3.3: Length
• Definition 3.4: Last passage value
• Remark