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arxiv: 2604.03358 · v1 · submitted 2026-04-03 · 🧮 math.PR

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

Pantelis Tassopoulos , Sourav Sarkar This is my paper

Pith reviewed 2026-05-13 18:02 UTC · model grok-4.3

classification 🧮 math.PR
keywords KPZ fixed pointabsolute continuityBrownian motionAiry sheetHausdorff dimensionhitting probabilitiesrecord timesthermal capacity
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The pith

The KPZ fixed point and Brownian motion share the same null sets on compact intervals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that increments of the KPZ fixed point, starting from any initial data, are mutually absolutely continuous to Brownian motion with diffusion constant 2 when restricted to any compact time interval. This means the two processes assign probability zero to exactly the same sets of paths on those intervals. The result extends an earlier one-sided absolute continuity statement and immediately transfers many path properties, such as certain hitting times and dimension calculations, between the two objects. It also shows a one-way absolute continuity between additive Brownian motion and the centred Airy sheet on compacts, while ruling out mutual absolute continuity on the whole line.

Core claim

The increments of the KPZ fixed point started from arbitrary initial data are mutually absolutely continuous with respect to Brownian motion with diffusion parameter 2 on compacts. Additive Brownian motion is absolutely continuous with respect to the centred Airy sheet on compacts, but the two are not mutually absolutely continuous globally. These relations imply that the KPZ fixed point and Brownian motion have identical null sets on every compact interval.

What carries the argument

Mutual absolute continuity of the law of KPZ fixed point increments and the law of Brownian motion with diffusion 2, on every compact time interval.

If this is right

  • With probability strictly between zero and one, the KPZ fixed point has record times away from any fixed reference point.
  • The probability that the graph of the KPZ fixed point hits a given set is positive precisely when that set has positive thermal capacity.
  • The essential supremum of the Hausdorff dimension of certain random intersections involving the KPZ fixed point can be computed explicitly.
  • Essential suprema of Hausdorff dimensions of images of plane sets under the Airy sheet are determined, along with a Bessel-Riesz capacity condition for positive Lebesgue measure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The local equivalence suggests that any null-set statement proved for Brownian motion on an interval can be imported directly into KPZ theory without additional work.
  • Global non-equivalence for the Airy sheet indicates that dimension and capacity results for the sheet may require separate global arguments even when local ones follow from Brownian motion.
  • The capacity characterization of hitting probabilities opens the possibility of using potential theory tools developed for Brownian motion to analyze KPZ level sets.

Load-bearing premise

The KPZ fixed point is well-defined as a continuous process for every choice of initial data.

What would settle it

A concrete compact time interval and a Borel set of paths with positive Lebesgue measure under Brownian motion but probability zero under the KPZ fixed point (or the reverse).

Figures

Figures reproduced from arXiv: 2604.03358 by Pantelis Tassopoulos, Sourav Sarkar.

Figure 1
Figure 1. Figure 1: Left: the centred KPZ fixed point started from flat initial data. Right: Brownian motion with diffusion parameter 2. Contents 1. Introduction 2 1.1. Related works 3 1.2. Organization of the paper 4 1.3. Notation 5 2. Preliminaries 5 2.1. Last passage percolation 5 2.2. Pitman transform 6 2.3. Brownian bridge properties 6 2.4. Airy line ensemble and the Brownian Gibbs property 7 2.5. The Airy sheet and the … view at source ↗
Figure 2
Figure 2. Figure 2: Flowchart of main steps in the proof of Theorem 3.7. we compute essential suprema of their Hausdorff dimensions of images of compact subsets in the plane under the Airy sheet and give a condition for them to have positive Lebesgue measure using potential theory for additive Brownian motion, see for example [Kho99]. 1.2. Organization of the paper. First, in Section 1.2 we provide necessary background materi… view at source ↗
Figure 3
Figure 3. 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 · δ[0,1] (recall (2.9)) (green) and the superposition of two narrow wedges at 0, 1, that is 0 · δ{0,1} (blue). We now prove a continuity result for the infinite last passage representation of the KPZ fixed point at the origin. This is the content of the following lem… view at source ↗
Figure 4
Figure 4. 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 (ε, A1(ε − A1(0) + G1), c… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the KPZ fixed point at unit time, h1(·, h0) on the compact interval [0, y0] in Theorem 3.7 on the event the top lines of the Pitman transforms in (3.4), F1(·)∨G1(·) does not hit F2(·)∨G2(·). On this event, h1(·, h0) is exactly equal to the Airy2 process up to a height shift that is G -measurable (recall the notation from Theorem 3.7. Moreover, the ‘lower barrier’ F2(·) ∨ G2(·) is G -measura… view at source ↗
Figure 6
Figure 6. Figure 6: Illustrations of: left: Additive Brownian motion, centre-left: S(·, · ′ ), centre-right: S(·, 0) + S(0, · ′ ) and right: S(·, · ′ ) − S(·, 0) − S(0, · ′ ) + S(0, 0) on [0, 1]2 . Observe the quadrangle inequality 2.6 S(x, y) + S(x ′ , y′ ) ≥ S(x, y′ ) + S(x ′ , y), x ≤ x ′ , y ≤ y ′ gives for any x0, y0, the random continuous function S(x, y) − S(x, y0) − S(x0, y) + S(x0, y0), x ≥ x0 , y ≥ y0 is monotone in… view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the graph of the rescaled increments of the KPZ fixed point in blue and the set E × F in red. The probability h(E) ∩ F 6= ∅ if and only if the graph of the KPZ fixed point, Gr(h) hits the red region with positive probability [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that increments of the KPZ fixed point started from arbitrary initial data are mutually absolutely continuous with respect to Brownian motion (diffusion parameter 2) on compact sets, extending the one-sided result of Sarkar et al. (2021). It further shows that additive Brownian motion is absolutely continuous w.r.t. the centred Airy sheet on compacts but not mutually globally, and derives applications including record times of the KPZ fixed point, a thermal-capacity characterization of hitting probabilities for the graph, essential suprema of Hausdorff dimensions of intersections, and Hausdorff dimensions of images under the Airy sheet together with a Bessel-Riesz capacity criterion for positive Lebesgue measure.

Significance. If the mutual absolute continuity on compacts holds for arbitrary initial data, the result supplies a robust mechanism for transferring null-set properties between the KPZ fixed point and Brownian motion, which would immediately strengthen fractal and capacity statements for the KPZ fixed point and the Airy sheet. The applications to record times, thermal capacity, and essential suprema of Hausdorff dimensions constitute concrete, falsifiable consequences that go beyond the abstract absolute-continuity statement.

major comments (1)
  1. [Abstract and main theorem statement] Abstract and main theorem: the claim of mutual absolute continuity on compacts for arbitrary initial data requires both KPZ increments ≪ BM and BM ≪ KPZ increments. While the forward direction follows from the cited Sarkar et al. (2021) one-sided result, the reverse direction for truly arbitrary (possibly non-continuous or super-linear) initial data is not shown to follow from a symmetry or approximation argument that avoids extra growth restrictions; the global non-mutuality noted for the Airy sheet indicates that the two directions are not symmetric and therefore cannot be taken for granted on compacts without an explicit construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for raising this important point about the scope of the mutual absolute continuity result. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract and main theorem statement] Abstract and main theorem: the claim of mutual absolute continuity on compacts for arbitrary initial data requires both KPZ increments ≪ BM and BM ≪ KPZ increments. While the forward direction follows from the cited Sarkar et al. (2021) one-sided result, the reverse direction for truly arbitrary (possibly non-continuous or super-linear) initial data is not shown to follow from a symmetry or approximation argument that avoids extra growth restrictions; the global non-mutuality noted for the Airy sheet indicates that the two directions are not symmetric and therefore cannot be taken for granted on compacts without an explicit construction.

    Authors: We thank the referee for this observation. The reverse direction is established explicitly in the proof of Theorem 1.1 via the variational formula for the KPZ fixed point, which is valid for arbitrary (including non-continuous and super-linear) initial data. On any compact interval the solution depends only on the initial data restricted to a slightly larger compact set; outside this set the initial data can be truncated or approximated by continuous functions with linear growth without changing the values on the compact, and the absolute continuity passes to the limit by tightness of the laws. This local character is what distinguishes the compact case from the global non-mutuality with the Airy sheet. We will add a short clarifying paragraph immediately after Theorem 1.1 to spell out this approximation step. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to one-sided prior result; extension adds independent content

full rationale

The paper extends the one-sided absolute continuity of KPZ fixed point increments with Brownian motion from the cited prior work sarkar2021brownian to a mutual version on compacts for arbitrary initial data. This self-citation is present in the abstract but is not load-bearing for the central claim, as the manuscript supplies new arguments for the reverse direction plus applications to record times, thermal capacity characterizations, and Hausdorff dimension computations. No step reduces a prediction or uniqueness claim to a fitted input, self-definition, or ansatz smuggled via citation; the derivation chain remains self-contained with independent mathematical content beyond the referenced one-sided relation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence and continuity of the KPZ fixed point for arbitrary initial data and on the one-sided absolute continuity result from the cited 2021 paper; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The KPZ fixed point exists as a continuous random function for arbitrary initial data.
    Invoked implicitly when stating the result for arbitrary initial data.
  • domain assumption The one-sided absolute continuity from sarkar2021brownian holds.
    The abstract states that the new result extends this prior theorem.

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Reference graph

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