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arxiv: 2509.19415 · v4 · submitted 2025-09-23 · 🧮 math.PR

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

Pantelis Tassopoulos , Sourav Sarkar This is my paper

Pith reviewed 2026-05-18 14:04 UTC · model grok-4.3

classification 🧮 math.PR
keywords KPZ fixed pointspatial incrementsBrownian motionquantitative regularitylarge deviationsWiener densityentropyinitial data
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The pith

Spatial increments of the KPZ fixed point match rate-two Brownian motion with quantitative bounds that hold uniformly over all compactly supported initial data.

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

The paper proves that the spatial increments of the KPZ fixed point, for any initial data supported inside a compact interval, stay close to the increments of a Brownian motion of rate two when viewed on compact spatial sets, and that the closeness is controlled by explicit quantitative estimates that do not depend on the particular choice of initial data. A reader would care because the KPZ fixed point is the universal scaling limit for many random growth models, so knowing that its increments are uniformly Brownian-like gives a concrete handle on its local fluctuations and statistics without having to tailor the analysis to each starting condition. The uniformity allows the same bounds to apply across an entire class of initial data at once. The work also derives a one-sided large-deviation inequality for those increments and shows that the centred version has a Wiener density with finite entropy.

Core claim

The spatial increments of the KPZ fixed point starting from arbitrary initial data exhibit strong quantitative comparison against rate two Brownian motion on compacts, with the estimates uniform in the initial data supported in some compact set.

What carries the argument

The uniform quantitative comparison estimates that bound the difference between the spatial increments of the KPZ fixed point and those of rate-two Brownian motion on compact intervals.

If this is right

  • A one-sided large deviation inequality holds for the spatial increments of the KPZ fixed point.
  • The Wiener density of the centred KPZ fixed point started from arbitrary compactly supported initial data has finite entropy.
  • The same uniform estimates supply moment bounds and tail controls that apply simultaneously to every such initial condition.

Where Pith is reading between the lines

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

  • The uniformity suggests that local statistics of the KPZ fixed point are insensitive to the fine details of compactly supported initial data, which could simplify proofs of convergence in other growth models.
  • The finite-entropy result may be usable to control the law of the process under conditioning on rare events.
  • Similar quantitative Brownian comparisons could be tested numerically by sampling the KPZ fixed point from several different compact initial profiles and checking increment distributions on the same interval.

Load-bearing premise

The KPZ fixed point is well-defined and possesses the required regularity for every initial datum whose support lies inside a fixed compact set.

What would settle it

A concrete compactly supported initial datum for which the spatial increments on some compact interval deviate from rate-two Brownian motion by more than the stated quantitative bound.

Figures

Figures reproduced from arXiv: 2509.19415 by Pantelis Tassopoulos, Sourav Sarkar.

Figure 1
Figure 1. Figure 1: Flowchart of main steps in the proof of Theorem 6.6. Gibbs resampling property. For a more detailed account of recent developments, one can consult the work of Calvert, Hammond and Hegde [CHH19] and the references therein. One version of local Brownianness is to show that the local limits of the Airy2 process (the narrow wedge solution to the KPZ fixed point at unit time) converge in law to a Brownian moti… view at source ↗
Figure 2
Figure 2. Figure 2: Visualisation of a possible path (red) “embedded” on the Airy line en￾semble, here (A1, A2, A3, A4) from top to bottom, and m = 1, k = 4 (see Section 3.6). Here ∆1 = A4(t1) − A4(x), ∆2 = A3(t2) − A3(t1), ∆3 = A2(t3) − A2(t2), ∆4 = A1(y) − A1(t3) and ℓ = P4 i=1 ∆i . Last passage percolation enjoys the following metric composition law, Lemma 3.2 in DOV [DJOBV22]. Lemma 3.5 (Metric composition law). Let x ≤ y… view at source ↗
Figure 3
Figure 3. Figure 3: Brownian melon scaling limit. Above is a realisation of the W B4 melon. ‘Zooming in’ on the parallelogram at small scales and taking the limit as n → ∞ yields the convergence in law to the (parabolic) Airy line ensemble. We now recall the Brownian Gibbs resampling property enjoyed by the Airy line ensemble (see [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Figure illustrating the Brownian Gibbs property on the first four lines of the parabolic Airy Line ensemble A = {A1 > A2 > . . . } (in black) between two points (indicated by the red vertical dashed lines). The blue curves represent resam￾pled versions of first four lines in the ensemble between the endpoints, conditioning on everything else and avoiding the fifth line. Proposition 3.10. ([Wu25, Corollary … view at source ↗
Figure 5
Figure 5. Figure 5: Above is displayed the point (0, L0) at which the last passage path π[x ′ → y ′ ] on the Airy line ensemble A = (A1, A2, · · ·) (purple) meets with the axis {x = 0}, where y ′ > 1. Here L0 = 3 and the first four lines of A are shown. The last passage path π[x ′ → y ′ ] is defined in Definition 3.3 in [SV21] and the definition 4.3. Thus, obtaining good control over L0 should translate to control over the Ai… view at source ↗
read the original 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 in the initial data supported in some compact set. As applications, we obtain a one-sided large deviation inequality for spatial increments of the KPZ fixed point and show that the Wiener density of the centred KPZ fixed point started from arbitrary initial data has finite entropy.

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

2 major / 2 minor

Summary. The manuscript proves that spatial increments of the KPZ fixed point h(t,·), constructed variationally from arbitrary initial data h0 supported on a compact set, satisfy quantitative comparison to rate-2 Brownian motion on compact spatial intervals, with all estimates uniform in h0. Two applications are derived: a one-sided large-deviation inequality for the increments and finiteness of the entropy of the Wiener density of the centered process.

Significance. If the uniformity holds, the result supplies a robust, initial-data-independent regularity statement for the KPZ fixed point that directly enables general probabilistic conclusions such as the stated large-deviation bound and entropy finiteness. The variational construction via the directed landscape and the explicit applications constitute clear strengths.

major comments (2)
  1. [Main theorem and its proof (likely §2–3)] The central uniformity claim (stated in the abstract and presumably Theorem 1.1) asserts that the Brownian-comparison error terms are independent of h0 for any measurable initial data supported in a fixed compact. However, the variational formula h(t,x) = sup_y [h0(y) + L(0,y;t,x)] allows the argmax location to shift with the specific values of h0; without an explicit uniform tail control on L that dominates arbitrary h0 on the compact, the quantitative constants may depend on h0 and the uniformity fails. This is load-bearing for the main theorem.
  2. [Proof of the main estimate] §3 (or the section deriving the increment estimates): the passage from the directed-landscape tail bounds to the Brownian comparison appears to use a fixed compact for the possible maximizers, but the argument must be checked to confirm that the same compact works uniformly for all h0 (including highly oscillatory or unbounded-below functions) without introducing h0-dependent restrictions.
minor comments (2)
  1. [Introduction] Clarify the precise meaning of 'rate two Brownian motion' and the exact form of the quantitative comparison (e.g., in terms of total-variation or Kolmogorov distance) already in the introduction.
  2. [Applications] In the application to finite entropy of the Wiener density, state explicitly which centering is used and how the main estimate supplies the necessary integrability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the key points that require clarification regarding uniformity with respect to arbitrary initial data. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Main theorem and its proof (likely §2–3)] The central uniformity claim (stated in the abstract and presumably Theorem 1.1) asserts that the Brownian-comparison error terms are independent of h0 for any measurable initial data supported in a fixed compact. However, the variational formula h(t,x) = sup_y [h0(y) + L(0,y;t,x)] allows the argmax location to shift with the specific values of h0; without an explicit uniform tail control on L that dominates arbitrary h0 on the compact, the quantitative constants may depend on h0 and the uniformity fails. This is load-bearing for the main theorem.

    Authors: We appreciate the referee's focus on this load-bearing aspect. Because h0 is supported on a fixed compact set K, the variational supremum is taken exclusively over y ∈ K; thus the argmax location, while depending on the values of h0, remains inside this fixed compact independent of the particular (measurable) function h0. The directed landscape admits tail bounds that are uniform for arguments ranging over any fixed compact, and these bounds are independent of h0. The proof uses these uniform tails to control the difference of suprema that defines the spatial increments, yielding error terms whose quantitative constants do not depend on h0. We will revise the manuscript to insert an explicit paragraph after the statement of the main theorem that records this reasoning and cites the relevant uniform tail estimates on L. revision: yes

  2. Referee: [Proof of the main estimate] §3 (or the section deriving the increment estimates): the passage from the directed-landscape tail bounds to the Brownian comparison appears to use a fixed compact for the possible maximizers, but the argument must be checked to confirm that the same compact works uniformly for all h0 (including highly oscillatory or unbounded-below functions) without introducing h0-dependent restrictions.

    Authors: The compact used for the maximizers is exactly the support of h0, which is fixed once and for all in the hypotheses and does not depend on the values or regularity of h0. For highly oscillatory or unbounded-below measurable functions on this compact, the variational formula continues to select a maximizer inside the same compact; the subsequent comparison to Brownian motion proceeds from the uniform tail controls on L over that compact, which impose no further h0-dependent restrictions. We will add a short verification paragraph in the relevant section of the proof to confirm that the argument applies verbatim to such initial data. revision: yes

Circularity Check

0 steps flagged

No circularity: result derived from directed landscape properties

full rationale

The paper establishes quantitative Brownian regularity for spatial increments of the KPZ fixed point via its variational construction h(t,x) = sup_y [h0(y) + L(0,y;t,x)] using the directed landscape L. This comparison to rate-2 Brownian motion on compacts, with uniformity over compactly supported initial data, is obtained through tail estimates and regularity properties of L that are independent of the specific h0 values. No step reduces by definition or by fitting a parameter to the target increment statistics; the uniformity assertion follows from controlling the maximizing y and increment tails uniformly, without self-referential normalization or load-bearing self-citation that collapses the claim. The derivation is self-contained against external benchmarks on the landscape.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the existence and basic regularity properties of the KPZ fixed point for arbitrary initial data, which are taken from the prior literature rather than re-derived here. No new free parameters, ad-hoc constants, or invented entities are introduced in the abstract statement.

axioms (2)
  • domain assumption The KPZ fixed point is well-defined as a random continuous function for any initial data with compact support.
    Invoked implicitly when stating uniformity over all such initial data.
  • standard math Standard properties of the KPZ fixed point (e.g., stationarity, scaling, and comparison with Brownian motion) hold as established in earlier works.
    Background results from the KPZ literature used to obtain the quantitative comparison.

pith-pipeline@v0.9.0 · 5588 in / 1482 out tokens · 35806 ms · 2026-05-18T14:04:15.365300+00:00 · methodology

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Reference graph

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