Invariant measures and shocks in the KPZ fixed point

Alexander Dunlap; Evan Sorensen

REVIEW 2 major objections 5 minor 16 references

Invariant measures and shocks in the KPZ fixed point

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that every time-ergodic invariant measure of the recentered KPZ fixed point is a two-sided Brownian motion with drift, and constructs a new invariant measure as seen from a shock.

desk verdict Full classification of extremal invariant measures for the KPZ fixed point, with new shock-frame measures and shock fluctuations; a serious paper that leans on one unverified lemma from a co-authored preprint. read the letter →

arxiv 2508.15598 v3 pith:IATFJEK5 submitted 2025-08-21 math.PR

Alexander Dunlap , Evan Sorensen This is my paper · ORCID

classification math.PR MSC 60K3560J65

keywords KPZfixedpointinvariantmeasuresdirectedlandscapeshockscompetitioninterfaceBessel-3processstationaryhorizonTracy-WidomGOE

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper attempts to finish the classification of the stationary states of the KPZ fixed point, the universal scaling limit of a broad class of random growth models. The central assertion is that, after subtracting the height at the origin, the only time-ergodic invariant measures are two-sided Brownian motions with drift; in particular, no V-shaped steady states exist. To account for the natural place where a shock lives, the authors construct a new family of invariant measures seen from the competition interface: the law of an independent Brownian motion plus a Bessel-3 process with drift 2θ. These shock-frame measures, they show, are exactly what emerges when one recenters the conjectural open KPZ fixed point stationary measures by their shock location in the L→∞ limit. The result would pin down the long-time statistics of shocks—diffusive in the stationary case, t^(1/3) Tracy–Widom GOE in the flat case—and would close the question of what the KPZ fixed point can look like in equilibrium.

What carries the argument

The load-bearing objects are three. First, the directed landscape L and the KPZ fixed point evolution h(x,t|f) = sup_y [f(y) + L(y,0;x,t)]; all invariance and fluctuation statements are proved through this variational formula. Second, the competition interface, defined through left and right interfaces I_p^± from the supremum difference d_p, locates the shock; for the shock-frame measure the interface is a single point b_t with probability one, and recentering by b_t is the correct shift. Third, the stationary horizon and the shock-frame measure bνθ, built from a Brownian motion paired with an independent process that is a Brownian motion with drift on one side and the 2M−X reflection (a Bes

What would settle it

Construct, exactly or numerically, a time-ergodic measure supported on V-shaped functions (lim f(x)/|x| = 2θ) that is invariant under the recentered KPZ fixed point; or simulate the KPZ fixed point from f(x) = 2θ|x| and observe whether the profile recentered at the competition interface stabilizes rather than spreading on the √t scale the paper predicts. A sharper check is to verify the preprint input directly: an eternal solution built from a V-shaped invariant measure would produce semi-infinite geodesics with more than three distinct directions, contradicting Lemma 4.3.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 1.3: a probability measure on the state space CFP is an extremal (time-ergodic) invariant measure for the recentered KPZ fixed point if and only if it is BM(2θ, √2), the law of a two-sided Brownian motion of diffusivity √2 and drift 2θ, for some real θ. The proof has two parts. First, using an eternal solution and the directional structure of semi-infinite geodesics, any extremal invariant measure must live on functions with deterministic asymptotic slopes at +∞ and −∞; the known uniqueness results for the stationary horizon then reduce the possibilities to drifted Brownian motions or V-shaped functions with slopes −2θ and +2θ. Second, the V

Load-bearing premise

The proof that every steady state has fixed slopes at ±∞ relies on an unrefereed preprint's claim that infinite-length paths traced backwards from the steady state always settle into a single direction; if that claim fails, V-shaped steady states could still exist.

Editorial extensions

If this is right

Every time-ergodic stationary state of the recentered KPZ fixed point is a two-sided Brownian motion with drift; there is no room for shock-like V-shaped steady states in the fixed frame.
The Brownian-plus-Bessel-3 shock-frame measure is invariant when viewed from the moving shock, so it is the correct stationary object for a competition interface in the KPZ scaling limit.
The open KPZ fixed point stationary measures, recentered at their shock, converge to the full-line shock-frame measure, with the shock's relative location becoming uniform on [0,1].
Shock fluctuations follow distinct universality classes: √t Gaussian fluctuations when the background is stationary, and t^(1/3) fluctuations with independent GOE Tracy–Widom marginals for flat deterministic V-shaped data.
The shock location from the shock-frame measure is diffusive, while the gap between left and right interfaces collapses, so the two interfaces merge in the scaling limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same classification mechanism should apply to zero-temperature models without integrable structure, such as Brownian last-passage percolation, where the only stationary states seen in the fixed frame are expected to be Brownian motions with drift.
The shock-frame measure is a plausible continuum analogue of the blocking measures that vanish from ASEP's extremal stationary measures; if so, it should appear as the environment seen from a second-class particle in the KPZ scaling limit.
The t^(1/3) shock fluctuation law for flat V-shaped data is likely universal beyond the KPZ fixed point and could be tested by measuring competition-interface locations in numerical simulations of the KPZ equation or in experimental growth systems.
The three-direction bound for eternal solutions suggests that higher-order multi-shock configurations cannot persist as stationary states, so any multi-shock structure must be transient rather than equilibrium.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—two things to know about this paper. First, it proves the full classification of extremal invariant measures for the recentered KPZ fixed point: only Brownian motions with drift. That was open for the V-shaped case, and prior work needed slope assumptions. Second, it constructs a new family of shock-frame invariant measures—Brownian plus independent Bessel-3—and derives shock fluctuation laws. These are genuinely new and not a repackaging of earlier results.

What the paper does well: the construction of the shock-frame measures is explicit and parameter-free; the invariance proofs are self-contained modulo standard facts about Bessel processes. The classification argument is well-organized: prove all extremal measures have deterministic asymptotic slopes, then rule out V-shaped measures via eternal solutions and geodesic directions. The paper does not hide its dependencies—it openly cites [BBS25] for the key lemma that eternal solutions generate semi-infinite geodesics, and the rest of the slope argument uses published results (BSS24a, Bus23). I read the proofs in some detail; the internal logic is coherent, and the appendices fill the gaps.

The soft spot is exactly that dependency. Proposition 4.4, and therefore the exclusion of V-shaped measures, rests on Proposition B.1 from [BBS25], a 2025 preprint co-authored by the second author. That lemma asserts that from any eternal solution satisfying the variational equality, the rightmost maximizers are semi-infinite geodesics. It is not proved here, and the paper does not verify any extra hypotheses beyond the existence of maximizers. If that lemma has a hidden flaw, the slope dichotomy collapses and the V-shaped case is left open. This is a genuine correctness risk, not a stylistic quibble. That said, the lemma is consistent with published results, and the paper flags the citation openly. It's the kind of risk that peer review is meant to resolve, not a reason to desk-reject.

Minor point: Theorem 1.2 concerns the conjectural open KPZ fixed point, but the convergence result is a real statement about the conjectural stationary measures, so I don't count that as a flaw.

Bottom line: this is a serious paper by people who know the area, and it deserves a rigorous referee. The referee should focus on the verification of Proposition B.1 in the context of Lemma 4.1, and on whether the eternal solution produced there actually satisfies the hypotheses. If that checks out, the classification is solid. Send it to peer review, and cite it if you work on KPZ.

Referee Report

2 major / 5 minor

Summary. The paper studies invariant measures and shock fluctuations for the KPZ fixed point, using the variational description via the directed landscape. It constructs a family of shock-frame invariant measures (Brownian motion plus an independent Bessel-3 process with drift 2θ), proves they arise as limits of the Barraquand–Corwin–Yang measures for the conjectural open KPZ fixed point after recentering by a shock location, classifies all extremal invariant measures for the recentered KPZ fixed point as two-sided drifted Brownian motions BM(2θ,√2), and derives distributional limits for shock locations in three regimes: √t fluctuations for the shock-frame and stationary-horizon initial data, and t^{1/3} Tracy–Widom GOE fluctuations for deterministic V-shaped initial data. The proofs rely heavily on the stationary horizon, eternal solutions, semi-infinite geodesics, and Bessel path-decomposition results.

Significance. If the central classification (Theorem 1.3) is correct, it resolves a natural open problem for the KPZ fixed point: the only extremal invariant measures are the known drifted Brownian motions, and there are no V-shaped extremal invariant measures. The shock-frame measure in Theorem 1.1 is a new explicit invariant object, and Theorem 1.4 gives sharp shock-fluctuation exponents with explicit limiting laws. The paper is careful and detailed: appendices contain substantial technical work on the enlarged state space CFP, Bessel processes, and Brownian meanders, and the main constructions have no fitted parameters. The main caveats are that the classification relies on an unproved lemma from a co-authored preprint [BBS25], and Theorem 1.2 concerns the conjectural open KPZ fixed point; the latter is clearly stated as conjectural in the abstract and introduction.

major comments (2)

[§4.1 / Appendix B, Proposition B.1] Proposition 4.4, the key slope-classification step for Theorem 1.3, imports [BBS25, Lemma A.12] as Proposition B.1 and uses it to assert that an eternal solution satisfying the variational equality gives rise to semi-infinite geodesics with deterministic directions. This imported lemma is not proved in the present paper, is not peer-reviewed, and is co-authored by the second author. The paper also does not verify the hypotheses of Proposition B.1 for the particular eternal solution b constructed in Lemma 4.1 beyond the variational equality (B.1) itself. Since Corollary 4.5 and the exclusion of V-shaped measures in §4.2 depend directly on Proposition 4.4, a gap or hidden assumption in Proposition B.1 would leave the V-shaped case unresolved and Theorem 1.3 unproved. The authors should either provide a complete proof of Proposition B.1 (or a self-contained verification of its hypotheses in
[§4.1, proof of Proposition 4.4] The proof derives that the right asymptotic slope is 2β and the left asymptotic slope is −2α, but at that point α and β are only described as 'possibly random'. The proposition statement requires deterministic constants, and Corollary 4.5 relies on this determinism. The missing step is an explicit use of extremality/ergodicity: the asymptotic slopes are conserved quantities (Lemma A.3), so under an extremal invariant measure they must be almost surely constant. This argument should be written out; as it stands, the proof appears to jump from random slopes to deterministic conclusions.

minor comments (5)

[§1.3] Typo: 'The prof of invariance' should be 'The proof of invariance'.
[Appendix D] Typo: 'standard Brownian mander' should be 'standard Brownian meander'.
[Lemma 4.3 proof] The sentence 'Dividing by |r| and taking limits as r → −∞' should read 'taking limits as r → ∞', since r>0 and t−r → −∞ as r→∞.
[§1.2 and Theorem 1.1] The notation BES3(2θ) is defined as a two-sided process via |B_d(x)+θx e1|, but this is easy to miss; for Theorem 1.1 it may help to explicitly say that Z is the two-sided Bessel process with the usual BES3(2θ) law on [0,∞) and the reflected/negative Bessel law on (−∞,0], as constructed in §3.2.
[Theorem 1.3] The parenthetical 'i.e., time-ergodic' is slightly imprecise: extremality in the convex set of invariant measures is equivalent to time-ergodicity for a Feller Markov process on a Polish space, but this equivalence is not stated. A brief remark would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new shock-frame measures are built from explicit Brownian/Bessel inputs and proven invariant; the classification leans on independent prior theorems, not on its own conclusion.

full rationale

The derivation chain is self-contained in the required sense: no target statement is fed back as an input. Theorem 1.1 defines f explicitly as B+Z (BM independent of BES3(2θ)) and proves shock-frame invariance by time-averaging a jointly invariant pair (Lemma 3.4, Theorem 3.5), using Rogers–Pitman path decomposition and Bessel facts; there is no fitted parameter renamed as a prediction. Theorem 1.2 identifies the shock as A=argmin Z of the BCY measure and proves convergence to the same B+Z law via the arcsine law, Denisov's decomposition, and meander-to-Bessel convergence; the location is natural, not chosen to force the conclusion. Theorem 1.3 rests on Proposition 4.4 (eternal solution implies deterministic slopes) and Proposition 2.11 (uniqueness under slope basins). The former imports [BBS25, Lemma A.12] and [BSS24a, Bus23] for semi-infinite geodesics and coalescence; these are prior theorems whose hypotheses (variational equality, existence of maximizers, directed landscape) do not include the classification, so the self-citations are load-bearing but not circular. The latter is a published, externally checkable uniqueness theorem ([BSS24a]); using it to force Brownian motions once slopes are known is a standard logical step, not an import of the conclusion. The new V-shape exclusion (Section 4.2) is an independent argument using tightness, Krylov–Bogoliubov, and the fluctuation non-tightness from Proposition 4.6. Two non-circular flags: [BBS25] is a co-authored unpublished preprint, so Proposition 4.4 carries a correctness risk if Lemma A.12 fails; and the proof of Proposition 4.4 cites "modulus of continuity bounds in Lemma A.1" although Lemma A.1 states symmetries (the needed bounds are in Lemma A.2/DOV22). Neither makes the argument circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; θ is a family label for the invariant measures, not an ad hoc constant. The new shock-frame measure bνθ is constructed explicitly from Brownian motion and Bessel-3 processes, and the shock location A in the open setting is the argmin of a Brownian functional, so no unexplained entities are introduced.

assumptions (6)

domain assumption Directed landscape L exists with metric composition, stationarity, skew-stationarity, rescaling, and growth bounds (Lemma A.1, A.2).
The KPZ fixed point is realized as sup_y f(y)+L(y,0;x,t); all evolution, geodesic, and invariance arguments in Sections 2-5 presuppose these properties from [DOV22] and [DV21].
domain assumption Initial data in the space CFP lead to continuous, non-exploding solutions and CFP is preserved (Proposition C.4).
Used to define the state space and to justify that invariant measures lie on continuous functions; proven in Appendix C from [MQR21] and directed landscape bounds.
domain assumption Stationary horizon invariance and uniqueness of invariant measures under slope conditions (1.6) (Propositions 2.10, 2.11).
Pulls in [BSS24a], which establishes the Busemann process and the fact that νθ is the unique invariant measure under slope conditions; used in Corollary 4.5, Proposition 4.8, and Theorem 1.2.
domain assumption For any eternal solution of the KPZ fixed point, leftmost/rightmost maximizers define semi-infinite geodesics (Proposition B.1, from [BBS25]).
Critical for Lemma 4.3 and Proposition 4.4 to show extremal invariant measures have deterministic slopes; [BBS25] is an unpublished preprint co-authored by the second author.
standard math Standard Brownian/Bessel path decompositions: Pitman-Rogers, Denisov's decomposition, arcsine law, and convergence of Brownian meanders to Bessel-3 (Appendix D).
Used in Theorem 1.2 and Lemma 3.4 to identify the shock-frame limit as Brownian plus Bessel-3.
standard math Flat-initial-data one-point law is Tracy-Widom GOE, and Airy_1 marginals are stationary with GOE distribution.
Used in Proposition 5.6 to obtain the t^{1/3} shock fluctuation for deterministic V-shaped data; cited to [MQR21], [FS05].

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Pith. "Pith review of Invariant measures and shocks in the KPZ fixed point." pith.science (2026). https://pith.science/paper/IATFJEK5

@misc{pith2026250815598,
  author       = {Pith},
  title        = {Pith review of: Invariant measures and shocks in the KPZ fixed point},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IATFJEK5}},
  note         = {Machine review of arXiv:2508.15598}
}

abstract

We construct a family of invariant measures from the perspective of a shock in the KPZ fixed point. These measures are parameterized by a positive number $\theta > 0$, and are supported on functions $f$ satisfying $\lim_{|x| \to \infty} \frac{f(x)}{|x|} = 2\theta$. Each can be described as the sum of a Brownian motion and an independent Bessel-$3$ process with drift. We show that these measures appear as the $L \to \infty$ limit of the (conjectural) stationary measures for the conjectural open KPZ fixed point on $[0,L]$, after recentering by an appropriately defined shock location. Furthermore, we show that, with respect to the standard, deterministic recentering at $x = 0$, all extremal invariant measures for the KPZ fixed point are Brownian motions with drift. To do this, we first show that any extremal invariant measures must be supported on functions having fixed asymptotic slopes at $\pm \infty$. Using a one-force-one-solution principle from the work of Busani, Sepp\"al\"ainen, and the second author, this rules out all other invariant measures except those having left slope $-2\theta$ and right slope $+2\theta$ for some $\theta > 0$. To handle this case, we derive the limiting fluctuations of the shock for a special choice of initial condition. Additionally, we derive the limiting fluctuations of the shock for the case of the invariant measure from the perspective of a shock, and for the case of initial data $f(x) = 2\theta|x|$.

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Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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KPZ fixed point convergence of the ASEP and stochastic six- vertex models

[ACH24a] Amol Aggarwal, Ivan Corwin, and Milind Hegde. KPZ fixed point convergence of the ASEP and stochastic six- vertex models. Preprint:arXiv:2412.18117,

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work page Pith review arXiv

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[Wu23] Xuan Wu. The KPZ equation and the directed landscape. Preprint arXiv:2301.00547 ,

arXiv