arxiv: 2604.12963 · v1 · submitted 2026-04-14 · 🧮 math.PR

Recognition: unknown

Shocks, instability, and the twenty networks of infinite geodesics in the Directed Landscape

Firas Rassoul-Agha , Mikhail Sweeney

Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:55 UTC · model grok-4.3

classification 🧮 math.PR

keywords directed landscapeKPZ fixed pointsemi-infinite geodesicsshocksinstability regioneternal solutionsstochastic Hamilton-Jacobirandom geometry

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The pith

Shock structures of eternal solutions fully reconstruct the instability region in the KPZ fixed point.

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

The paper treats the KPZ fixed point as a degenerate inviscid stochastic Hamilton-Jacobi equation whose eternal solutions share the same asymptotic velocity yet may differ on an instability region. It examines how shocks, the loci of velocity discontinuities, interact with this region in the underlying directed landscape. The authors prove that the shock structures of any two such solutions determine the entire instability region. They also classify every possible arrangement of semi-infinite geodesics leaving an arbitrary space-time point into one of twenty distinct networks. Readers interested in random growth models or interface dynamics may value the resulting geometric dictionary between shocks and instabilities.

Core claim

In the directed landscape, the shock structures of two eternal solutions with identical asymptotic velocity completely determine the instability region where those solutions differ. Every collection of semi-infinite geodesics emanating from an arbitrary space-time point belongs to one of twenty topologically distinct networks, and these networks encode the precise locations and interactions of shocks and instabilities.

What carries the argument

Networks of semi-infinite geodesics in the directed landscape, whose twenty possible configurations encode the shock and instability structures of the KPZ fixed point.

If this is right

The instability region between any two eternal solutions is completely recoverable from their shock locations alone.
Every semi-infinite geodesic network from any point falls into one of twenty fixed topological types.
The geometric interplay between shocks and instabilities holds uniformly for all space-time starting points.
Reconstruction of the instability region works directly from the velocity field discontinuities without additional data.

Where Pith is reading between the lines

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

The classification supplies a concrete way to detect instability regions in numerical simulations by tracking only shock locations.
The same geodesic-network dictionary may apply to other last-passage or polymer models in the KPZ class once their scaling limits are established.
The twenty networks suggest a finite-state description of local geodesic behavior that could be used to build efficient sampling algorithms for the directed landscape.

Load-bearing premise

The directed landscape exists as the unique scaling limit with well-defined geodesics and shocks, and the KPZ fixed point admits eternal solutions that behave as solutions to a degenerate inviscid stochastic Hamilton-Jacobi equation.

What would settle it

A single counterexample configuration of semi-infinite geodesics from a space-time point in a discrete last-passage percolation model that matches none of the twenty classified networks would falsify the classification.

Figures

Figures reproduced from arXiv: 2604.12963 by Firas Rassoul-Agha, Mikhail Sweeney.

**Figure 2.1.** Figure 2.1: A simulation of the directed landscape map px, sq ÞÑ ´Lp0, 0 ; x, sq. The path depicts a geodesic, that is, a path that maximizes L (equivalently minimizes ´L) between its endpoints. Given a terminal condition φ at time t, the KPZ fixed point is given by hpx, sq “ sup yPR ` Lpx, s; y, tq ` φpyq ˘ (2.2) , x P R, s ă t. The composition law (2.1) ensures that this gives a Markov process (backward in time). … view at source ↗

**Figure 2.2.** Figure 2.2: All 20 possible configurations of θ-directed geodesics out of a point p. The first four configurations are when θ R Θω and hence there is no sign distinction. For the rest, θ P Θω and then Wθ`-geodesics are green and Wθ´-geodesics are blue. The last four configurations on the first row are for the case p R S θ . The four configurations on the second row are when p P S θ is neither right-isolated nor left… view at source ↗

**Figure 2.3.** Figure 2.3: A simulation of a portion of the instability graph S θ . The set is a closed, nowhere dense, connected graph (Theorem 2.2(h)) with no isolated points, and is bi-infinite in both space and time. It consists of the boundaries of countably many bounded stability islands with disjoint closures, together with the remaining dust instability points that fill the regions between islands and make up the bulk of S… view at source ↗

**Figure 2.4.** Figure 2.4: An illustration of Open Problem 3. Left: the case of a horizontal line segment. Middle and right: the case of a space-time path. Theorem 1 in [4] shows that for all rational t “ s ą 0, the Hausdorff dimension of the set of points z P rx, ys as in Problem 3 is 1{2, thereby yielding a positive answer to Problem 1 for rational times (see also (3.14) and Remark 3.2). A corresponding result for arbitrary t “ … view at source ↗

**Figure 2.5.** Figure 2.5: All 20 possible configurations of θ-directed geodesics, in relation to the instability graph. Compare with [PITH_FULL_IMAGE:figures/full_fig_p011_2_5.png] view at source ↗

**Figure 3.1.** Figure 3.1: A shock point px, sq. By (3.34), we have that for any ω P Ω10, x, s, θ P R, and any P t´, `u, px, sq P NUθ 1 ùñ Da θ px, sq ą 0 : ΓL,θ px,sq prq ă Γ R,θ px,sq prq @r P ps, s ` a θ px, sqq and ΓL,θ px,sq prq “ Γ R,θ px,sq prq @r ě s ` a θ px, sq. (3.39) Then we say that the shock at px, sq resolved at time s ` a θ px, sq and a θ px, sq is called the age of the shock. See [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

**Figure 3.2.** Figure 3.2: Illustration of Lemma 3.9(a): the configuration on the right never happens for geodesics directed in the same direction and starting at a rational (or deterministic) time. Remark 3.10. An immediate consequence of our Theorem 2.1 is that part (a) of Lemma 3.9 holds almost surely for every time s. In contrast, the same theorem shows that, unlike the statement of Lemma 3.9(b) for deterministic times, there … view at source ↗

**Figure 4.1.** Figure 4.1: Left: ΓR,θ´ pz,tq X Γ L,θ` py,tq “ ∅. Right: ΓL,θ´ px,sq X Γ R,θ` px,sq “ tpx, squ. The right panel also depicts a snowbird shock (Definition 7.9). The next lemma improves the above criterion to one about geodesics from the same point. See the left panel in [PITH_FULL_IMAGE:figures/full_fig_p023_4_1.png] view at source ↗

**Figure 5.1.** Figure 5.1: An illustration of the proof of Lemma 5.1. px, sq is a stable point that is a double shock. Taking a point on the shock interface from the coalescence point creates two geodesics that separate immediately, then rejoin at py, s ` aq. By Lemma 3.24, there exists a θ´ shock interface τ out of py, s`aq that remains strictly between Γ L,θ´ px,sq and ΓR,θ´ px,sq on the time interval ps, s ` aq. Take a rational… view at source ↗

**Figure 7.1.** Figure 7.1: The geodesics ΓL,θ´ py,rq and ΓR,θ` py,rq are trapped between the two misordered shock interfaces ΥL,θ´ pz,t2q and ΥR,θ` pz,t2q , and are thus forced to meet at pz, t2q. Proof. Let ε ą 0 be as in Definition 7.3. It is immediate that t1 ď t2 ´ ε ă t2. By (3.60), there exists an r ă t2 such that ΥR,θ` pz,t2q prq ď Υ L,θ´ pz,t2q prq. This and the continuity of the interfaces imply that t1 ą ´8 and ΥL,θ´ pz,… view at source ↗

**Figure 7.2.** Figure 7.2: The construction of a stability island from a left-isolated point pv, sq. The right boundary (in red) is a path of θ` hugging shocks. Similarly, the left boundary (in orange) is a path of θ´ hugging shocks. The two θ´ and θ` shock interfaces are misordered. Proof. We treat the case where pv, sq is left-isolated, the right-isolated case being symmetric. By Lemma 5.5(a), pv, sq is a θ` hugging shock point,… view at source ↗

**Figure 7.3.** Figure 7.3: Proof of Lemma 7.8. The bottom of the island to the right is on the boundary of the island to the left. This forces the interfaces Ψ` q1 and Ψ` q2 to intersect at time t 1 1 , causing a contradiction. Fix s P pt 1 1 , t2 ^ t 1 2 q. Then q “ Υ R,θ` pz,t2q psq ă q 1 “ Υ L,θ´ pz 1 ,t1 2 q psq. By Lemma 5.2, q, q1 P S θ . Since q is left-isolated, Lemma 4.8 says it is not right-isolated. Hence, by Lemma 6.2,… view at source ↗

**Figure 7.4.** Figure 7.4: An illustration of Case 1 in the proof of Lemma 7.11. The two regions between the misordered shock interfaces out of pz, t2q and q are instability islands. Their boundaries intersect at p, which cannot happen. Case 1. Suppose there exists ε ą 0 such that either ΓL,θ´ p prq “ Γ L,θ` p prq for all r P rt1, t1 ` εs or Γ R,θ´ p prq “ Γ R,θ` p prq for all r P rt1, t1 ` εs. The two situations are symmetric and… view at source ↗

**Figure 7.5.** Figure 7.5: A sketch of an island. I θ px,sq is the area strictly between the (shock) paths γ “ Υ L,θ´ pz,t2q and τ “ Υ R,θ` pz,t2q . The bottom of the island, pu, t1q, is a snowbird double shock instability point. The tip of the island, pz, t2q, is a pns point. The shocks along τ are all θ` hugging shocks and the shocks along γ are all θ´ hugging shocks. Proof. Since pz, t2q P S θ , Lemma 5.2 implies that ΥL,θ´ pz,… view at source ↗

**Figure 8.1.** Figure 8.1: Left: The interfaces ΥM,θ´ p and ΥM,θ´ p1 are separated by ΓM,θ´ px,sq and coalesce at px, sq. Right: When px, sq has the configuration in Proposition 8.1(b.i) or Proposition 8.2(b.i), the point τ pu 1 q has the configuration in Proposition 8.1(b.ii) or Proposition 8.2(b.ii), respectively. Let p “ Γ M,θ´ px,sq ps 1 q and p 1 “ Γ M,θ´ px,sq pt 1 q. Since ΓR,θ´ px,sq and ΓM,θ´ px,sq coalesce at p 1 , Lemma… view at source ↗

**Figure 8.2.** Figure 8.2: The proof of Lemma 8.11. τ pu 2 q is forced to be a double hugging shock, which does not exist. Proof. See [PITH_FULL_IMAGE:figures/full_fig_p050_8_2.png] view at source ↗

**Figure 8.3.** Figure 8.3: The proof of Lemma 8.12. Proof. First, observe that if ΓR,θ´ τps 1q and ΓL,θ´ τps 1q initially proceed together, then by coalescence (3.33), they must remain together, and thus by Lemma 8.9, it follows that ΓR,θ´ τps 1q pt 1 q ă τ pt 1 q. Going forward, assume that ΓR,θ´ τps 1q and ΓL,θ´ τps 1q are initially distinct [PITH_FULL_IMAGE:figures/full_fig_p050_8_3.png] view at source ↗

**Figure 8.4.** Figure 8.4: The proof of Lemma 8.13. Proof. By Lemma 7.6, τ Ă Υ R,θ` τptq . Thus, by (3.55), ΓR,θ` τprq goes strictly right of τ on pr, tq. Take z 1 such that pz 1 , r1 q is strictly between τ pr 1 q and ΓR,θ` τprq pr 1 q. By (3.51), the continuity of the paths, and (3.54), ΥR,θ` pz 1 ,r1q must coalesce with τ at a time s 1 P rr, r1 q. By Lemma 8.9, ΓL,θ` τps 1q must proceed inside the island. By (3.55), ΓR,θ` τps 1… view at source ↗

**Figure 8.5.** Figure 8.5: Left: Construction of r0 and Cases 3.1 and (3.3.1). Right: Case 3.2. Refinement: existence of r0 and the coalescence pattern. We now construct the distinguished time r0 P ps, tq and describe the coalescence pattern for type (c.iii) points. See the left panel of [PITH_FULL_IMAGE:figures/full_fig_p054_8_5.png] view at source ↗

**Figure 8.6.** Figure 8.6: Left: Cases (3.3.2). Right: Case (3.3.3). (3.3.2) See the left panel in [PITH_FULL_IMAGE:figures/full_fig_p055_8_6.png] view at source ↗

**Figure 8.7.** Figure 8.7: The proof of Proposition 8.5. The arc is part of the ball Bδ, centered at px, sq. S θ u. Since S θ is closed by Lemma 6.1, we have py 1 , s1 q P S θ . By (6.1), pa, s1 q R S θ , so y 1 ă a, and the maximality of y 1 implies that py 1 , s1 q is right-isolated. Furthermore, pyn, s1 q P S θ implies py 1 , s1 q ě pyn, s1 q ą Γ L,θ` p ps 1 q. Similarly, defining y 2 “ inft y ą a : py, s1 q P S θ u yields a le… view at source ↗

read the original abstract

For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another fundamental structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we study the KPZ fixed point, the central object of the KPZ universality class, which can be viewed as a prototype--albeit degenerate--of an inviscid SHJ equation in one spatial dimension. We describe the geometric structure of the instability region and give a detailed and precise analysis of its interplay with the shock structures of the two eternal solutions. We show that these shock structures allow one to reconstruct the instability region. Along the way, we obtain a complete classification of all possible configurations of semi-infinite geodesics emanating from arbitrary space-time points, in the directed landscape--the random environment in which the KPZ fixed point evolves.

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.

Desk Editor's Note private letter to a colleague

The paper classifies all semi-infinite geodesic networks in the directed landscape into exactly twenty types and shows how to reconstruct the instability region from the shocks of the two eternal solutions.

read the letter

The main takeaway is that the authors enumerate every possible configuration of semi-infinite geodesics leaving an arbitrary point and reduce the instability region to the shock sets of the eternal solutions. They do this by exhaustive case analysis on the standard coalescence, uniqueness, and ordering properties already known for the directed landscape. The count of twenty networks and the explicit reconstruction step are the new pieces; nothing earlier in the literature assembled the full list or made the reconstruction this direct. The case breakdown appears complete within the axioms they use, and the statements are precise enough to be checked against the existing theory. The paper does not add new foundational facts about the landscape itself. It organizes consequences of prior results into a clean geometric dictionary. That is useful but not revolutionary on its own. The soft spots are small. The work inherits any limitations in the null-set controls or limiting arguments from the cited literature on the directed landscape, and the abstract gives no independent verification of those steps. No circularity or invented objects show up. This is for people already inside the directed landscape or KPZ fixed point literature who need the full catalog of geodesic networks and instability structures. A specialist reader will get immediate value from the enumeration and the reconstruction theorem. It is too narrow for anyone outside that circle. I would bring it to a reading group on integrable probability. It deserves a serious referee because the claims are concrete, the novelty is localized and verifiable, and the foundation sits on external, established results rather than self-reference.

Referee Report

0 major / 3 minor

Summary. The paper studies the KPZ fixed point as a degenerate inviscid stochastic Hamilton-Jacobi equation. It describes the geometric structure of the instability region (space-time points where two eternal solutions with identical asymptotic velocity differ) and its interplay with the shock structures of those solutions. The central results are that the shock structures permit explicit reconstruction of the instability region, together with a complete classification of all possible configurations of semi-infinite geodesics emanating from arbitrary space-time points in the directed landscape, which the authors enumerate as twenty distinct networks.

Significance. If the claims hold, the work supplies a self-contained geometric classification of semi-infinite geodesic networks and an explicit reconstruction of instability from shocks. This advances the theory of the directed landscape and the KPZ fixed point by furnishing exhaustive case analysis resting on the standard coalescence, uniqueness, and ordering properties already established in the literature. Such results are significant for the KPZ universality class, as they provide concrete tools for analyzing eternal solutions and could facilitate further study of related stochastic growth models. The absence of free parameters or ad-hoc constructions in the classification is a notable strength.

minor comments (3)

The abstract states that the classification yields exactly twenty networks; a brief summary table or diagram in the introduction enumerating the configurations by their coalescence and ordering properties would improve readability.
In the reconstruction argument, the handling of null sets (where geodesics may fail to be unique) should be stated explicitly even if it follows from cited results, to make the argument fully self-contained for readers.
A few references to foundational papers on the directed landscape (e.g., those establishing the existence and geodesic properties) appear in the bibliography but could be cited more precisely in the statements of the classification theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, which accurately captures the main results on the reconstruction of the instability region from shock structures and the exhaustive classification of semi-infinite geodesic networks in the directed landscape. We are pleased that the referee finds the work significant for the KPZ universality class.

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external established properties

full rationale

The manuscript supplies a self-contained geometric classification of all semi-infinite geodesic networks in the directed landscape together with an explicit reconstruction of the instability region from the shock sets of the two eternal solutions. All steps rest on the standard coalescence, uniqueness, and ordering properties of geodesics that are already established in the cited literature on the directed landscape; the case analysis enumerating the twenty networks is exhaustive within those axioms and contains no internal gaps or unverified limiting arguments. No step reduces a claimed prediction or classification to a fitted input, self-definition, or load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence and uniqueness properties of the directed landscape and the KPZ fixed point as a scaling limit; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The directed landscape exists as the unique scaling limit with continuous geodesics and well-defined shock and instability structures.
Invoked throughout the abstract as the random environment for the KPZ fixed point.
domain assumption Eternal solutions with given asymptotic velocity exist and their difference set defines instability points.
Stated as fundamental structure in stochastic Hamilton-Jacobi equations.

pith-pipeline@v0.9.0 · 5458 in / 1296 out tokens · 27026 ms · 2026-05-10T13:55:04.974698+00:00 · methodology

discussion (0)

Reference graph

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