Pith. sign in

REVIEW 3 major objections 5 minor 75 references

This paper constructs the periodic directed landscape, proves it unique in law, and shows exponential LPP and periodic ASEP converge to it, giving a universal scaling limit for KPZ models on a ring.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 15:59 UTC pith:QMFJBP23

load-bearing objection Major periodic-KPZ construction with a concrete gap in the central Cauchy argument that needs a fix before the existence claim is solid. the 3 major comments →

arxiv 2607.18104 v1 pith:QMFJBP23 submitted 2026-07-20 math.PR math-phmath.MP

Periodic directed landscape

Amol Aggarwal , Ivan Corwin , Dominik Schmid This is my paper
classification math.PR math-phmath.MP MSC 60K3582C2260B10
keywords periodic directed landscapeKPZ universality classdirected landscape gluingKPZ fixed pointlast passage percolationasymmetric simple exclusion processvariational formulaperiodic geometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that a natural random object, the periodic directed landscape, exists and is unique in law on a cylinder geometry, and that it is the universal scaling limit for periodic models in the KPZ universality class. It proves that periodic exponential last passage percolation converges to this landscape, and that the periodic ASEP, under the basic coupling, converges to periodic KPZ fixed points coupled through the same landscape. The paper also identifies the periodic KPZ fixed point of Baik–Liao–Liu with a variational formula involving the periodic directed landscape. The central tool is a gluing construction that patches together full-space directed landscapes on short time intervals, which is of independent interest and opens the door to scaling limits in other geometries.

Core claim

The paper's central claim is the construction and characterization of a random continuous function L^per on (T×R)^2_↑, called the periodic directed landscape, which is the unique (in law) continuous random function satisfying three properties: local agreement with the full-space directed landscape, independent increments over disjoint time intervals, and a metric composition law of the form L^per(x,s;y,t) = sup_z [L^per(x,s;z,u) + L^per(z,u;y,t)]. The paper proves this object is the scaling limit of periodic exponential LPP, that it gives the variational representation of the periodic KPZ fixed point, and that coupled periodic ASEPs converge to coupled periodic KPZ fixed points driven by thi

What carries the argument

The core mechanism is a gluing construction that builds the periodic landscape from full-space directed landscapes. For each small time interval, two full-space landscapes are coupled to agree (with high probability) on short, well-separated rectangles in space-time (Lemma 2.8), then the landscapes are patched together via a metric composition law that propagates the values across time steps. The construction yields a dyadic sequence of 'patched directed landscapes' whose laws form a Cauchy sequence in a weighted total-variation metric; the periodic directed landscape is the unique limit. The proof relies on quantitative control of geodesic transversal fluctuations (e.g., Proposition 2.5) to

Load-bearing premise

The entire gluing construction rests on Lemma 2.8, which asserts that two full-space directed landscapes can be coupled to agree on specific short, well-separated rectangles with probability at least 1 − C exp(−c n²); if this exponential rate fails, the union bound over time steps breaks and the patched landscapes may not converge.

What would settle it

Compute (or rigorously bound) the total-variation distance between the restrictions of two independent full-space directed landscapes to the rectangles O^{i,1}_n and O^{i,3}_n at separation scale n^{2/3}; if the distance decays slower than exp(−c n²), the gluing construction collapses. A concrete test would be a numerical simulation of two independent directed landscapes on rectangles of width 1/4 and height 1/n for large n, measuring the probability that their values agree after optimal coupling, and checking whether this probability is consistent with an exp(−c n²) lower bound.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The variational formula h^per(h0;y,t) = sup_x [h0(x) + L^per(x,0;y,t)] provides a continuous Markov process description of the periodic KPZ fixed point, resolving the Baik–Liao–Liu conjecture.
  • Convergence of periodic ASEP to the same periodic directed landscape yields the q-independence of the scaling limit: any limit theorem known for periodic TASEP now holds for periodic ASEP.
  • Local agreement of L^per with the full-space directed landscape implies the periodic KPZ fixed point approximates the full-space KPZ fixed point at small times, confirming the BL24 conjecture on periodic-to-full-space convergence.
  • The construction extends to general period length p by scaling, giving the p-periodic directed landscape for any p>0.
  • The gluing technique is model-agnostic and can be reused: any full-space model known to converge to the directed landscape (or KPZ fixed point) can be converted into a periodic limiting object, provided one has appropriate geodesic or second-class-particle fluctuation bounds.

Where Pith is reading between the lines

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

  • The gluing framework presented here should apply directly to other full-space KPZ models, such as the KPZ stochastic PDE, the log-gamma polymer, or the stochastic six-vertex model, to produce their periodic scaling limits with only modest additional estimates.
  • The same construction could define a 'directed landscape' on more general space-time manifolds, including intervals and surfaces with boundary, by chopping the domain into full-space (or half-space) patches and gluing them back together.
  • The exponential-in-n^2 coupling rate is likely not sharp; if a slower rate were sufficient, the approach might extend to settings where full-space convergence holds with only polynomial error probabilities.
  • The periodic directed landscape provides a natural random directed metric on the torus; studying its geodesics and their winding behavior may lead to rigorous mixing-time results for periodic ASEP, as the paper suggests.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper constructs a random field L^per on (T×R)^2_↑, called the periodic directed landscape, by gluing restrictions of full-space directed landscapes on short space-time rectangles. It proves that this field is the unique-in-law continuous random function satisfying local agreement with the full-space directed landscape, independent increments on disjoint time intervals, and the metric composition law (Theorem 2.20). It then proves that periodic exponential last passage percolation converges to L^per (Theorem 3.18), identifies the Baik–Liao–Liu periodic KPZ fixed point with the variational formula sup_x[h_0^per(x)+L^per(x,0;y,t)] (Theorem 4.5), and proves that periodic ASEPs under the basic coupling converge jointly to periodic KPZ fixed points coupled through the same L^per (Theorem 5.9). The central tool is a gluing procedure for full-space directed landscapes and their prelimits, developed in Section 2 and used for all three convergence theorems.

Significance. If the construction is valid, this is a major advance: it supplies the conjectured periodic scaling limit for the KPZ universality class, proves its uniqueness and universality across periodic LPP and periodic ASEP, and resolves conjectures from BLL26 and BL24. The paper is also methodologically valuable: the gluing framework is modular, deliberately avoids circular dependence between the existence result and the convergence proofs, and uses a hierarchy of quantitative estimates (geodesic fluctuation bounds, moderate deviations, second-class-particle estimates). The explicit care to prove the LPP convergence (Theorem 3.18) without using the continuity or characterization theorems is a real strength, as is the transparency about deferred proofs in the appendices. However, the existence step contains a specific gap that must be repaired before the surrounding results can be accepted.

major comments (3)
  1. [§2.4, Eq. (2.87)] The Cauchy-sequence proof in Proposition 2.18 is not valid as written. The displayed chain bounds W*(ν_{2^n},ν_{2^ℓ}) by 2^{-n+2} + Σ_{i=n}^{ℓ-1} Σ_{j=1}^{i+1} 2^{-j} C exp(-c i^{-1/16}). But exp(-c i^{-1/16}) tends to 1 as i→∞, not to 0. The double sum is therefore at least c(ℓ-n) for large ℓ, which diverges; the final bound C_0 exp(-c_0 n^{-1/16}) does not follow. Since Proposition 2.18 is the step that produces the limit object L^per, this is load-bearing. The likely repair is to substitute the actual rate from Lemma 2.16, C exp(-c δ^{-1/32}), with a δ matched to the index j in the definition of W* (or to the set D_{δ_j}); the proof as written does not perform this matching.
  2. [§2.4, Theorem 2.20 (uniqueness)] The uniqueness argument is terse in a load-bearing spot: after Eq. (2.96), the proof asserts that Lemma 2.15 and Lemma 2.16 can be iterated to obtain the same geodesic-fluctuation control for the limiting objects L^per and \tilde L^per. Since L^per is obtained as a limit point in W*, not as a pathwise limit of the patched landscapes, this iteration requires a Borel–Cantelli argument along the dyadic coupling sequence. The intended argument is plausible, but it should be written out, because the uniqueness conclusion ('any random field satisfying (i'), (ii), (iii) must be a periodic directed landscape') depends on it.
  3. [§1.2 and Theorem B] The informal Theorem B states uniqueness under (1) local agreement, (2) independent increments, and (3) metric composition. The precise uniqueness statement, Theorem 2.20(i'), requires a quantitative local-agreement rate α>2 and restricts to continuous fields. Without the rate, the announced 'unique' characterization is not what is proved. The introduction should either state the quantitative assumptions or note that the informal statement is a compressed version of Theorem 2.20.
minor comments (5)
  1. [§2.4, Eq. (2.84)] The notation δ_n = n^{-1/16} is introduced just before Proposition 2.18, but in the proof the same symbol δ appears to be reused for a different scale in the application of Lemma 2.16. Please disambiguate the two scales; this would also make the error in Eq. (2.87) easier to spot.
  2. [§2.1, Lemma 2.8] The proof of Lemma 2.8 cites HP24 Lemma 3.2 'on the event X ≤ 1/8 n^{2/3}'. After the scale change, the separation between the rectangles is of order n^{2/3}. A short sentence explaining why the failure probability becomes C exp(-c n^2) would help the reader verify that the imported estimate is used with the correct exponent.
  3. [§2.4, Lemma 2.17] The proof of completeness of (M(C_per_up), W*) is shortened by an appeal to 'Proposition 6 in [GS84]' for completeness in total variation. Since W* is a weighted sum of total-variation distances, this is fine, but a direct proof would be more self-contained.
  4. [§3.5, Theorem 3.18] The theorem states convergence 'in the sense of uniform convergence on compact sets'. Because the periodic directed landscape is only known to be upper semi-continuous at that point in the argument, the topology on the space of upper semi-continuous functions should be specified. This is a presentation issue, not a mathematical one, since Proposition 2.19 later upgrades the limit to continuous.
  5. [Appendix A and B] Several load-bearing moderate-deviation estimates (Lemmas 3.9, 3.10, 3.14, Proposition 3.12, and the second-class-particle bounds in Section 5) are deferred. The paper is long, but a brief summary of the proof strategy for at least Lemmas 3.9 and 3.10 in the main text would improve readability.

Circularity Check

0 steps flagged

No circularity: gluing construction and convergence proofs reduce to external full-space benchmarks, not to the target claims.

full rationale

The paper's derivation chain is not circular. The periodic directed landscape is constructed by gluing full-space directed landscapes, with the load-bearing coupling Lemma 2.8 imported from external approximate-independence results ([HP24], [Dau24]) rather than from the target object. The convergence of periodic LPP (Theorem 3.18) is proved by coupling periodic LPP to full-space LPP and to the patched landscapes, not by assuming the limit. The identification of the periodic KPZ fixed point (Theorem 4.5) is a genuine convergence proof: periodic TASEP is embedded in periodic LPP and the limit is shown to be sup_x h0(x)+L^per, with the BLL26 formulas used only to state the target, not as an input to the proof. The periodic ASEP convergence (Theorem 5.9) uses full-space ASEP convergence from [ACH24a] and patched couplings; this is an external (though same-author) benchmark with independent content, so it does not amount to self-citation circularity. The uniqueness/characterization Theorem 2.20 is proved within the paper rather than imported from prior work. No fitted parameter is renamed as a prediction. The most serious issue I see is non-circular: the dyadic Cauchy bound in Proposition 2.18, inequality (2.87), appears to use a rate exp(-c i^{-1/16}) that does not decay as i grows, which would make the displayed chain fail; this is a correctness gap in the existence argument, not a reduction-by-construction, and therefore does not raise the circularity score.

Axiom & Free-Parameter Ledger

2 free parameters · 10 axioms · 3 invented entities

The central claim rests on the full-space directed landscape and its quantitative properties ([DOV22], [DV21], [GZ22], [HP24], [Dau24], [ACH24a]) plus self-contained but deferred moderate-deviation estimates for periodic LPP (Appendix A) and ASEP second-class particles (Appendix B, via [FS25]). No numbers are fitted to data anywhere; the only hand-chosen constants are convenience exponents (1/16 in δ_n) and geometry constants (1/8, 1/16). The main domain restrictions are the filling-fraction bound (3.32) and the Bernoulli-sandwiching assumption (5.20).

free parameters (2)
  • dyadic patch-scale exponent 1/16 (δ_n = n^{−1/16}, eq. (2.84)) = 1/16
    Hand-chosen convenience exponent in the exhaustive slope sequence D_{δ_n}. The text itself says it 'may be improved'. Not fitted to data and not load-bearing: any sufficiently small exponent would serve.
  • gluing geometry constants (1/8, 3/8, 5/8, 9/8, 1/16, 1/32) = rational constants (1/8, 1/16, etc.)
    Hand-chosen spacing and widths of the rectangles O^{i,j}_n and R^{i,j}_n in (2.19) and (2.25). Any sufficiently separated, sufficiently overlapping intervals would serve; the 1/32 tolerance in Lemma 2.7 fixes the 1/8 scale. Chosen by hand to make the derivation work, but no fitting to data.
axioms (10)
  • standard math Existence, uniqueness, and basic properties of the full-space directed landscape (Definition 2.1; local Airy sheet structure, metric composition, geodesics, modulus of continuity).
    Invoked throughout Section 2 as the base object; imported from [DOV22] (Theorems 10.9, 12.1, Corollary 10.7, Proposition 1.6). This is the external benchmark the gluing construction builds on.
  • standard math Full-space exponential LPP converges to the directed landscape under KPZ scaling (Theorem 3.5).
    Imported from [DV21, Theorem 1.7 and Remark 1.10]; used in Lemma 3.17 to transfer full-space LPP convergence to patched landscapes.
  • standard math Approximate independence of the directed landscape on distant, short rectangles (Lemma 2.8).
    Key input to the gluing: agreement probability 1 − C exp(−c n²) on O^{i,1}_n, O^{i,2}_n, O^{i,3}_n. Imported from Lemma 3.2 of [HP24] and Proposition 2.6 of [Dau24]; used in the union bound (2.28).
  • standard math Geodesic transversal-fluctuation bounds for the full-space directed landscape (Proposition 2.5).
    Imported from [GZ22, Lemma 3.11]; exponent 9/4 controls Lemma 2.7 (ε-good geodesics), Lemma 2.10, and Lemma 2.15.
  • standard math Full-space coupled ASEP converges to coupled KPZ fixed points (Theorem 5.6).
    Imported from [ACH24a, Theorem 2.10]; the input that lets the patched ASEP converge to patched KPZ fixed points in Lemma 5.23.
  • standard math Moderate deviations for periodic LPP geodesic transversal fluctuations (Lemmas 3.9, 3.10, 3.14; Proposition 3.12).
    Proved in Appendix A, which is not visible in the provided text. Theorem 3.18 rests on them; flagged as unverifiable in this review.
  • domain assumption Second-class particle moderate-deviation bounds for the (extended) ASEP couplings (Proposition 5.14).
    The paper states these have origins in the recent work [FS25]; proof outlined in Appendix B. Load-bearing for the reference-frame coupling that drives the periodic ASEP convergence proof.
  • domain assumption Filling fraction non-degeneracy: a ≤ lim inf k/N ≤ lim sup k/N ≤ 1−a for some a > 0 (eq. (3.32)).
    Uniform bound on the particle density needed for the periodic moderate deviations; a genuine restriction on the models covered by Theorem 3.18.
  • domain assumption Initial particle configurations sandwiched stochastically between Bernoulli(ρ ± C N^{−1/2}) measures (eq. (5.20)).
    Assumption on initial data for Theorem 5.9 (periodic ASEP convergence). The authors flag in Remark 5.10 that it likely can be relaxed considerably.
  • domain assumption The periodic KPZ fixed point of [BLL26] exists as the TASEP scaling limit with explicit finite-dimensional distribution formulas (Theorem 4.2).
    Inherited from [BLL26]; the object this paper identifies with the variational formula. The resolution of the conjecture is tied to the equation numbering of the first arXiv posting of [BLL26].
invented entities (3)
  • Periodic directed landscape L^per independent evidence
    purpose: Conjectured universal scaling limit for KPZ models in periodic geometry; the main new object of the paper.
    Constructed as the unique W*-limit of patched directed landscapes (Proposition 2.18), characterized uniquely by locality + independence + metric composition (Theorem 2.20), and given falsifiable handles: local agreement with the full-space directed landscape (2.88), convergence of periodic LPP (Theorem 3.18), and convergence of periodic ASEP (Theorem 5.9).
  • Unwrapped/wrapped patched directed landscapes L̄_n, L_n independent evidence
    purpose: Finite-scale approximations built by gluing coupled full-space directed landscapes; intermediate construction objects.
    Explicitly defined (Definition 2.9) in terms of the coupled full-space landscapes, with quantified agreement probabilities (Lemmas 2.10-2.16). Every stated property is checkable from the construction, and any failure would be detectable in the error estimates.
  • Patched KPZ fixed point h^{(n)} independent evidence
    purpose: Intermediate object for the ASEP convergence proof (Definition 5.22).
    Defined explicitly via the variational formula using the patched landscape; used only as an approximation intermediate whose total-variation distance to both the patched ASEP and the periodic KPZ fixed point is quantified (Corollary 5.21, Proposition 2.18).

pith-pipeline@v1.3.0-alltime-deepseek · 74193 in / 24431 out tokens · 241670 ms · 2026-08-01T15:59:58.011557+00:00 · methodology

0 comments
read the original abstract

We construct the periodic directed landscape, which is the conjectured scaling limit for periodic models in the Kardar-Parisi-Zhang universality class. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape. Moreover, we confirm conjectures by Baik, Liao and Liu on the local structure of the periodic KPZ fixed point, and establish the convergence of periodic ASEPs to periodic KPZ fixed points, coupled according to the same periodic directed landscape. Our main tool to construct the periodic directed landscape (and prove convergence to it) is a technique for gluing full-space directed landscapes (and their prelimits), which is of independent interest.

Figures

Figures reproduced from arXiv: 2607.18104 by Amol Aggarwal, Dominik Schmid, Ivan Corwin.

Figure 1
Figure 1. Figure 1: Models in a periodic geometry: (a) depicts periodic ASEP where particles (black) jump subject to exclusion with rates q and 1 as illustrated; (b) depicts the mapping between ASEP and a height function (−1 slope increments enter sites occupied by particles and +1 slope increments enter empty sites); (c) illustrates the characteristic KPZ scaling underwhich we show that ASEP and other models have a scaling l… view at source ↗
Figure 2
Figure 2. Figure 2: An example of a time-varying domain for last passage percolation. Initially (time is measured in the diagonal direction) the domain is periodic but at some time it splits into two periodic domains and then later one of them (on the upper branch) opens into a model on a strip with different rates on the two sides, and finally the two branches join together again. The last passage time optimizes the sum of w… view at source ↗
Figure 3
Figure 3. Figure 3: An approximation scheme for the periodic heat equation. construct the periodic directed landscape. To show that this scheme indeed works, we will need to use probabilistic versions of finite speed of propagation estimates, in the form of geodesic fluctuation bounds or, in the case of ASEP, second-class particle fluctuation bounds. We illustrate our gluing approach first in the context of solving the period… view at source ↗
Figure 4
Figure 4. Figure 4: The gluing construction for the patched directed landscape uses the coupling Pper of full-space directed landscapes L 1 and L 2 from Lemma 2.8. Under that coupling, L 1 restricted to O i,1 n agrees with L 2 restricted to O i,3 n and L 1 and L 2 restricted to O i,2 n agrees with each other. exact same manner as for the bulk time increments. If there is no bulk, i.e. k < 2, then the argument needs only to be… view at source ↗
Figure 5
Figure 5. Figure 5: Assignment of the different rectangles in the construction of patched directed landscape. Note that horizontal coordinates x, y are at least 1 16 away from the right boundary of R i,1 n , and that we assign Rx,s;y,t = R i,1 n in this exam￾ple. In particular, when the coupling from Lemma 2.8 is successful, we get that the directed landscapes L 1 and L 2 agree on O i,1 n = R i,1 n ∩ Ri,2 n . Definition 2.9. … view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the subset R∗ = Re1 ∩ R1,1 n \ (O 1,1 n ∪ O1,2 n ) ⊆ Re of the rectangle Re marked in purple. for ¯π n,− x,s;y,t. Note that by (2.44), the path ¯π n,+ x,s;y,t is indeed an inherited geodesic for L¯ n, and hence by construction the rightmost such path. A similar argument holds for ¯π n,− x,s;y,t. This gives (2.40) for all (x, s; y, t) ∈ (R×[0, 1])2 ↑ ∩ Dn−1 . The corresponding result for the… view at source ↗
Figure 7
Figure 7. Figure 7: Path decomposition for π from (x, s) to (y, t) with arrival times (t π i )i∈J3K and departure times (s π i )i∈J3K . of these parts separately. More precisely, for the first step, note that without loss of generality, for every path π connecting two points (x, s) to (y, t), we can decompose π into finitely many parts according to the times (t π k )k≥1 (respectively (s π k )k≥1) such that we first (last) hit… view at source ↗
Figure 8
Figure 8. Figure 8: Path decomposition for π on the event A (2) δ . The dashed vertical lines in red are at least 1 8 apart while the dashed horizontal lines in purple are at heights zi−1 and zi for some i, respectively, so that ℓ, ℓ′ ∈ [zi−1, zi ]. P [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Rectangles R i,j n and R¯i,j n used in the construction of patched last passage percolation for ρ = 1, rotated by π/4. We denote by rℓ(i, j) and ¯rℓ(i, j) for ℓ ∈ J4K the corners of the respective rectangles. Note that the path πu,v is an element of V N n by bounding the distance between u and v perpendicular to the (1, 1)-direction. The next result states that with high probability, all lattice paths in a… view at source ↗
Figure 10
Figure 10. Figure 10: Assignment of the different rectangles used in the construction of patched last passage percolation for ρ = 1, rotated by π/4. In Lemma 3.13, we argue that whenever the geodesics πr˜1(i,j),r˜2(i,j) and πr˜4(i,j+1),r˜3(i,j+1) do not leave the rectangles R i,j n \ R¯i,j n and R i,j+1 n \ R¯i,j+1 n , respectively, then the last passage times can be coupled to agree on the light purple shaded part R¯i,j n ∩ R… view at source ↗
Figure 11
Figure 11. Figure 11: Visualization of the correspondence between the periodic TASEP and periodic last passage percolation at time 0 (left) and at some time T > 0 (right). At the right-hand side, all shaded sites below the red curve have a last passage time of at most T to the black curve, while all sites above the red curve have a last passage time larger than T to the black curve. The red curve’s vertical / horizontal increm… view at source ↗
Figure 12
Figure 12. Figure 12: Visualization of the ASEP on Z with jump attempts at times t and s. The particle attempting the jump is drawn in blue. • B = 1: Then, if η(x) = 1−η(x+ 1) = 1 holds, we move the particle at site x to site x+ 1 in configuration η. Similarly, if η ′ (x) = 1 − η ′ (x + 1) = 1 holds, we move the particle at site x to site x + 1 in configuration η ′ . • B = 0. Then, if η(x) = 1 − η(x + 1) = 0 holds, we move the… view at source ↗
Figure 13
Figure 13. Figure 13: Construction of the ASEP (ηt)t≥0 on Z with blue arrows, and the unwrapped periodic ASEP (¯η per t )t≥0 under the (ρ, θ, 1, N)-reference frame basic coupling with θ = 0, N = 4 and some ρ < 1 2 shaded in green. Arrows indicate the Poisson clocks along the edges, while the direction of the arrows indicate the outcome of the Bernoulli-(1 + q) −1 -random variables. Purple arrows belong to both processes, while… view at source ↗
Figure 14
Figure 14. Figure 14: Visualization of the extended colored ASEP (ξt)t≥0 from Defini￾tion 5.13 with respect to (ηt)t≥0 and (˜ηt)t≥0 on Z. with X ∼ ν and X′ ∼ ν ′ . We have the following comparison between (ηt)t≥0 and (¯η per t )t≥0 evolving according to the extended (ρ, θ, 1, N)-reference frame basic coupling. Proposition 5.14. Let C > 0, θ ∈ R, N ∈ N and a ∈ (0, 1). Let ρ ∈ [a, 1 − a] and consider ASEPs (ηt)t≥0 and (¯η per t … view at source ↗
Figure 15
Figure 15. Figure 15: Visualization of the path decomposition in the proof of Lemma A.3 in the case v2 − u2 > 4n −1/3N3/2 . The path γ per u,v is sandwiched between the paths γ per u′ ,w− and γ per u′ ,w+ between the levels (i − 1)n −1N3/2 and in−1N3/2 . Since (u, v) ∈ (Xn ∩ Xn ) ∩ DN (ℓn)−1 by our assumptions, the event in (A.25) ensures that ∆˜ [PITH_FULL_IMAGE:figures/full_fig_p079_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Visualization of the path sandwiching in the proof of Proposi￾tion 3.12. The set Ext(u −, u+, v+, v−) is depicted as purple dots. on an event of probability at least 1 − exp(−c0n 1/6 ) for all n ∈ N large enough. A visualization of this construction in given in [PITH_FULL_IMAGE:figures/full_fig_p081_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Visualization of the site u − and u + and the ball B¯N δ around u used in the proof of Lemma 3.14, where we see that the geodesic from u to v is sandwiched between the geodesics from u − to v and from u + to v, respectively. Define the points w − := (γu−,v(u − 2 + δ 1/4N 3/2 ), u− 2 + δ 1/4N 3/2 ), w+ := (γu+,v(u + 2 + δ 1/4N 3/2 ), u− 2 + δ 1/4N 3/2 ) lying on the geodesics γu−,v and γu+,v, respectively.… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

75 extracted references · 13 linked inside Pith

  1. [1]

    Colored line ensembles for stochastic vertex models

    Amol Aggarwal and Alexei Borodin. Colored line ensembles for stochastic vertex models. Selecta Mathematica , 30(5):105, Nov 2024

    2024

  2. [2]

    The ASEP speed process

    Amol Aggarwal, Ivan Corwin, and Promit Ghosal. The ASEP speed process . Advances in Mathematics , 422:109004, 2022

    2022

  3. [3]

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

    Amol Aggarwal, Ivan Corwin, and Milind Hegde. KPZ fixed point convergence of the ASEP and stochastic six-vertex models . Preprint, https://arxiv.org/abs/2412.18117 , 2024

    Pith/arXiv arXiv 2024

  4. [4]

    Scaling limit of the colored ASEP and stochastic six-vertex models

    Amol Aggarwal, Ivan Corwin, and Milind Hegde. Scaling limit of the colored ASEP and stochastic six-vertex models . Preprint, https://arxiv.org/abs/2403.01341 , 2024

    Pith/arXiv arXiv 2024

  5. [5]

    Strong characterization for the A iry line ensemble

    Amol Aggarwal and Jiaoyang Huang. Strong characterization for the A iry line ensemble. Invent. Math. , 243(2):293--605, 2026

    2026

  6. [6]

    Fluctuations for the T oda lattice

    Amol Aggarwal and Matthew Nicoletti. Fluctuations for the T oda lattice. Preprint, https://arxiv.org/abs/2604.14346 , 2026

    Pith/arXiv arXiv 2026

  7. [7]

    KPZ limit theorems

    Jinho Baik. KPZ limit theorems . In Proceedings of the 2022 International Congress of Mathematicians . EMS Press, 2022

    2022

  8. [8]

    Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

    Guillaume Barraquand, Alexei Borodin, Ivan Corwin, and Michael Wheeler. Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Mathematical Journal , 167(13):2457–2529, 2018

    2018

  9. [9]

    The stochastic heat equation: Feynman-Kac formula and intermittence

    Lorenzo Bertini and Nicoletta Cancrini. The stochastic heat equation: Feynman-Kac formula and intermittence . Journal of Statistical Physics , 78(5):1377--1401, Mar 1995

    1995

  10. [10]

    On the distribution of the length of the longest increasing subsequence of random permutations

    Jinho Baik, Percy Deift, and Kurt Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. Journal of the American Mathematical Society , 12(4):1119--1178, 1999

    1999

  11. [11]

    Stochastic Burgers and KPZ Equations from Particle Systems

    Lorenzo Bertini and Giambattista Giacomin. Stochastic Burgers and KPZ Equations from Particle Systems . Communications in Mathematical Physics , 183(3):571--607, Feb 1997

    1997

  12. [12]

    Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

    Erik Bates, Shirshendu Ganguly, and Alan Hammond. Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape. Electronic Journal of Probability , 27:1--44, 2022

    2022

  13. [13]

    Bahadoran, H

    C. Bahadoran, H. Guiol, K. Ravishankar, and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems . The Annals of Probability , 34(4):1339 -- 1369, 2006

    2006

  14. [14]

    Nonexistence of bigeodesics in integrable models of last passage percolation

    Riddhipratim Basu, Christopher Hoffman, and Allan Sly. Nonexistence of bigeodesics in integrable models of last passage percolation. Communications in Mathematical Physics , pages 1--30, 2021

    2021

  15. [15]

    TASEP on a ring in sub-relaxation time scale

    Jinho Baik and Zhipeng Liu. TASEP on a ring in sub-relaxation time scale . Journal of Statistical Physics , 165(6):1051–1085, 2016

    2016

  16. [16]

    Fluctuations of TASEP on a ring in relaxation time scale

    Jinho Baik and Zhipeng Liu. Fluctuations of TASEP on a ring in relaxation time scale . Communications on Pure and Applied Mathematics , 71(4):747–813, 2018

    2018

  17. [17]

    Multipoint distribution of periodic TASEP

    Jinho Baik and Zhipeng Liu. Multipoint distribution of periodic TASEP . Journal of the American Mathematical Society , 32(3):609–674, 2019

    2019

  18. [18]

    Periodic TASEP with general initial conditions

    Jinho Baik and Zhipeng Liu. Periodic TASEP with general initial conditions . Probability Theory and Related Fields , 179(3-4):1047–1144, 2021

    2021

  19. [19]

    Pinched-up periodic KPZ fixed point

    Jinho Baik and Zhipeng Liu. Pinched-up periodic KPZ fixed point . Preprint, https://arxiv.org/abs/2403.01624 , 2024

    Pith/arXiv arXiv 2024

  20. [20]

    Large Time Cumulants of the KPZ Equation on an Interval: G

    Guillaume Barraquand and Pierre Le Doussal. Large Time Cumulants of the KPZ Equation on an Interval: G. Barraquand, P. Le Doussal . Journal of Statistical Physics , 192(8):111, 2025

    2025

  21. [21]

    Periodic KPZ fixed point with general initial conditions

    Jinho Baik, Yuchen Liao, and Zhipeng Liu. Periodic KPZ fixed point with general initial conditions . Preprint, https://arxiv.org/abs/2603.01964 , 2026

    Pith/arXiv arXiv 2026

  22. [22]

    Limiting one-point distribution of periodic TASEP

    Jinho Baik, Zhipeng Liu, and Guilherme LF Silva. Limiting one-point distribution of periodic TASEP . Annales de l'Institut henri poincare (B) probabilites et statistiques , 58:248–302, 2022

    2022

  23. [23]

    Coalescence of geodesics in exactly solvable models of last passage percolation

    Riddhipratim Basu, Sourav Sarkar, and Allan Sly. Coalescence of geodesics in exactly solvable models of last passage percolation. Journal of Mathematical Physics , 60(9):093301, 22, 2019

    2019

  24. [24]

    Periodic Pitman Transforms and Jointly Invariant Measures

    Ivan Corwin, Yu Gu, and Evan Sorensen. Periodic Pitman Transforms and Jointly Invariant Measures . Communications in Mathematical Physics , 407(3):49, Feb 2026

    2026

  25. [25]

    The Kardar-Parisi-Zhang equation and universality class

    Ivan Corwin. The Kardar-Parisi-Zhang equation and universality class . Random Matrices Theory Appl. , 1(1):1130001, 76, 2012

    2012

  26. [26]

    Non-uniqueness times for the maximizer of the KPZ fixed point

    Duncan Dauvergne. Non-uniqueness times for the maximizer of the KPZ fixed point . Advances in Mathematics , 442:109550, 2024

    2024

  27. [27]

    Jan de Gier, Richard Kenyon, and Samuel S. Watson. Limit shapes for the asymmetric five vertex model. Communications in Mathematical Physics , 385(2):793--836, Jul 2021

    2021

  28. [28]

    The directed landscape

    Duncan Dauvergne, Janosch Ortmann, and Bálint Virág. The directed landscape. Acta Mathematica , 229(2):201–285, 2022

    2022

  29. [29]

    The scaling limit of the longest increasing subsequence

    Duncan Dauvergne and Bálint Virág. The scaling limit of the longest increasing subsequence. Preprint, https://arxiv.org/abs/2104.08210 , 2021

    Pith/arXiv arXiv 2021

  30. [30]

    The directed landscape from Brownian motion

    Duncan Dauvergne and Balint Virag. The directed landscape from Brownian motion . Preprint, https://arxiv.org/abs/2405.00194 , 2024

    Pith/arXiv arXiv 2024

  31. [31]

    Characterization of the directed landscape from the KPZ fixed point

    Duncan Dauvergne and Lingfu Zhang. Characterization of the directed landscape from the KPZ fixed point . Preprint, https://arxiv.org/abs/2412.13032 , 2024

    Pith/arXiv arXiv 2024

  32. [32]

    The directed landscape in half-space

    Duncan Dauvergne and Lingfu Zhang. The directed landscape in half-space. Preprint, https://arxiv.org/abs/2604.10020 , 2026

    Pith/arXiv arXiv 2026

  33. [33]

    Khanin, A

    Weinan E, K. Khanin, A. Mazel, and Ya. Sinai. Invariant Measures for Burgers Equation with Stochastic Forcing . Annals of Mathematics , 151(3):877--960, 2000

    2000

  34. [34]

    Mixing times and cutoff for the TASEP in the high and low density phase

    Dor Elboim and Dominik Schmid. Mixing times and cutoff for the TASEP in the high and low density phase . Probability and Mathematical Physics , 5(2):413–459, 2024

    2024

  35. [35]

    Mixing times for the open ASEP at the triple point

    Patrik L Ferrari and Dominik Schmid. Mixing times for the open ASEP at the triple point . Preprint, https://arxiv.org/abs/2501.03951 , 2025

    Pith/arXiv arXiv 2025

  36. [36]

    Some recent progress on the periodic KPZ equation

    Yu Gu and Tomasz Komorowski. Some recent progress on the periodic KPZ equation . Stochastics and Partial Differential Equations: Analysis and Computations , Jan 2025

    2025

  37. [37]

    Solutions in the large for nonlinear hyperbolic systems of equations

    James Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. , 18:697--715, 1965

    1965

  38. [38]

    Mixing times for the simple exclusion process with open boundaries

    Nina Gantert, Evita Nestoridi, and Dominik Schmid. Mixing times for the simple exclusion process with open boundaries. Annals of Applied Probability , 33(2):1172–1212, 2023

    2023

  39. [39]

    Givens and Rae Michael Shortt

    Clark R. Givens and Rae Michael Shortt. A class of Wasserstein metrics for probability distributions. Michigan Mathematical Journal , 31(2):231 -- 240, 1984

    1984

  40. [40]

    Six-vertex model, roughened surfaces, and an asymmetric spin hamiltonian

    Leh-Hun Gwa and Herbert Spohn. Six-vertex model, roughened surfaces, and an asymmetric spin hamiltonian. Phys. Rev. Lett. , 68:725--728, Feb 1992

    1992

  41. [41]

    Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times

    Shirshendu Ganguly and Lingfu Zhang. Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times. Preprint, https://arxiv.org/abs/2204.01674 , 2022

    Pith/arXiv arXiv 2022

  42. [42]

    A patchwork quilt sewn from brownian fabric: Regularity of polymer weight profiles in brownian last passage percolation

    Alan Hammond. A patchwork quilt sewn from brownian fabric: Regularity of polymer weight profiles in brownian last passage percolation. Forum of Mathematics, Pi , 7:e2, 2019

    2019

  43. [43]

    Boundary current fluctuations for the half space ASEP and six vertex model

    Jimmy He. Boundary current fluctuations for the half space ASEP and six vertex model . Proceedings of the London Mathematical Society , 128(2):e12585, 2024

    2024

  44. [44]

    The directed landscape is a black noise

    Zoe Himwich and Shalin Parekh. The directed landscape is a black noise. preprint, https://arxiv.org/abs/2404.16801 , 2024

    Pith/arXiv arXiv 2024

  45. [45]

    Shape fluctuations and random matrices

    Kurt Johansson. Shape fluctuations and random matrices. Communications in Mathematical Physics , 209(2):437–476, 2000

    2000

  46. [46]

    Discrete polynuclear growth and determinantal processes

    Kurt Johansson. Discrete polynuclear growth and determinantal processes. Communications in Mathematical Physics , 242(1-2):277--329, 2003

    2003

  47. [47]

    Dynamic scaling of growing interfaces

    Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. , 56:889--892, Mar 1986

    1986

  48. [48]

    Multi-point distribution of discrete time periodic tasep

    Yuchen Liao. Multi-point distribution of discrete time periodic tasep. Probability Theory and Related Fields , 182(3):1053--1131, Apr 2022

    2022

  49. [49]

    Elliott H. Lieb. Residual entropy of square ice. Phys. Rev. , 162:162--172, Oct 1967

    1967

  50. [50]

    Thomas M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. Transactions of the American Mathematical Society , 213:237--261, 1975

    1975

  51. [51]

    Thomas M. Liggett. Interacting Particle Systems . Classics in mathematics. Springer-Verlag, Berlin, 2005

    2005

  52. [52]

    Multipoint distribution of TASEP

    Zhipeng Liu. Multipoint distribution of TASEP . The Annals of Probability , 50(4):1255--1321, 2022

    2022

  53. [53]

    Small deviations for beta ensembles

    Michel Ledoux and Brian Rider. Small deviations for beta ensembles. Electronic Journal of Probability , 15:no. 41, 1319–1343, 2010

    2010

  54. [54]

    Contour integral formulas for PushASEP on the ring

    Jhih-Huang Li and Axel Saenz. Contour integral formulas for PushASEP on the ring . The Annals of Probability , 53(4):1434 -- 1490, 2025

    2025

  55. [55]

    Integral Formulas of ASEP and q-TAZRP on a Ring

    Zhipeng Liu, Axel Saenz, and Dong Wang. Integral Formulas of ASEP and q-TAZRP on a Ring . Communications in Mathematical Physics , 379(1):261--325, Oct 2020

    2020

  56. [56]

    Melenk and I

    J.M. Melenk and I. Babuška. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering , 139(1):289--314, 1996

    1996

  57. [57]

    MacDonald, Julian H

    Carolyn T. MacDonald, Julian H. Gibbs, and Allen C. Pipkin. Kinetics of biopolymerization on nucleic acid templates. Biopolymers , 6(1):1--25, 1968

    1968

  58. [58]

    The KPZ fixed point

    Konstantin Matetski, Jeremy Quastel, and Daniel Remenik. The KPZ fixed point . Acta Mathematica , 227(1):115--203, 2021

    2021

  59. [59]

    One-sided reflected Brownian motions and the KPZ fixed point

    Mihai Nica, Jeremy Quastel, and Daniel Remenik. One-sided reflected Brownian motions and the KPZ fixed point . In Forum of Mathematics, Sigma , volume 8, page e63. Cambridge University Press, 2020

    2020

  60. [60]

    Current fluctuations and large deviations for periodic tasep on the relaxation scale

    Sylvain Prolhac. Current fluctuations and large deviations for periodic tasep on the relaxation scale. Journal of Statistical Mechanics: Theory and Experiment , 2015(11):P11028, nov 2015

    2015

  61. [61]

    Riemann surface for TASEP with periodic boundaries

    Sylvain Prolhac. Riemann surface for TASEP with periodic boundaries . Journal of Physics A: Mathematical and Theoretical , 53(44):445003, oct 2020

    2020

  62. [62]

    Scale invariance of the PNG droplet and the A iry process

    Michael Pr\"ahofer and Herbert Spohn. Scale invariance of the PNG droplet and the A iry process. Journal of Statistical Physics , 108(5--6):1071--1106, 2002

    2002

  63. [63]

    Can extra updates delay mixing? Communications in Mathematical Physics , 323(3):1007--1016, 2013

    Yuval Peres and Peter Winkler. Can extra updates delay mixing? Communications in Mathematical Physics , 323(3):1007--1016, 2013

    2013

  64. [64]

    The One-Dimensional KPZ Equation and Its Universality Class

    Jeremy Quastel and Herbert Spohn. The One-Dimensional KPZ Equation and Its Universality Class . Journal of Statistical Physics , 160(4):965--984, Aug 2015

    2015

  65. [65]

    Nonequilibrium behaviour of a many particle process: density profile and local equilibria

    Herman Rost. Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete , 58(1):41–53, 1981

    1981

  66. [66]

    Beta ensembles, stochastic Airy spectrum, and a diffusion

    Jose Ramirez, Brian Rider, and B \'a lint Vir \'a g. Beta ensembles, stochastic Airy spectrum, and a diffusion . Journal of the American Mathematical Society , 24(4):919--944, 2011

    2011

  67. [67]

    Sch \"u tz

    Gunter M. Sch \"u tz. Exact solution of the master equation for the asymmetric exclusion process. Journal of Statistical Physics , 88(1):427--445, Jul 1997

    1997

  68. [68]

    Mixing times for the TASEP in the maximal current phase

    Dominik Schmid. Mixing times for the TASEP in the maximal current phase . Annals of Probability , 51(4):1342--1379, 2023

    2023

  69. [69]

    Systems of conservations laws 1

    Denis Serre. Systems of conservations laws 1 . Cambridge University Press, 1999

    1999

  70. [70]

    Interaction of Markov processes

    Frank Spitzer. Interaction of Markov processes . Advances in Mathematics , 5(2):246--290, 1970

    1970

  71. [71]

    Mixing times for the TASEP on the circle

    Dominik Schmid and Allan Sly. Mixing times for the TASEP on the circle . Probability Theory and Related Fields , pages 1--73, 2026

    2026

  72. [72]

    Tracy and Harold Widom

    Craig A. Tracy and Harold Widom. Asymptotics in ASEP with step initial condition. Communications in Mathematical Physics , 290(1):129--154, 2009

    2009

  73. [73]

    The KPZ equation and the directed landscape

    Xuan Wu. The KPZ equation and the directed landscape . The Annals of Probability , 54(3):1212--1257, 2026

    2026

  74. [74]

    TASEP in half-space

    Xincheng Zhang. TASEP in half-space . Preprint, https://arxiv.org/abs/2409.09974 , 2025

    Pith/arXiv arXiv 2025

  75. [75]

    Convergence from the Log-Gamma Polymer to the Directed Landscape

    Xinyi Zhang. Convergence from the Log-Gamma Polymer to the Directed Landscape . Preprint, https://arxiv.org/abs/2505.05685 , 2025

    arXiv 2025