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arxiv: 2002.09459 · v5 · submitted 2020-02-21 · 🧮 math.PR · math.CO

Hidden invariance of last passage percolation and directed polymers

Duncan Dauvergne This is my paper

Pith reviewed 2026-05-24 15:25 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords last passage percolationdirected polymersRSK correspondencegeometric RSKshift invariancerearrangement invarianceKPZ equationAiry sheet
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The pith

Integrable last passage percolation and directed polymer models gain shift and rearrangement invariances from RSK structure and decoupling.

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

The paper shows that last passage percolation and directed polymer models on the integer lattice are invariant under translations and certain reflections. When these models also admit an integrable structure via the RSK correspondence or geometric RSK correspondence, such as in geometric last passage percolation or the log-gamma polymer, the basic invariances combine with a decoupling property to produce additional symmetries. The resulting symmetries include shift and rearrangement invariance for last passage times, geodesic locations, disjointness probabilities, polymer partition functions, and quenched polymer measures. The same framework yields scrambled versions of the classical RSK correspondence and an RSK correspondence for moon polyominoes, with the statements carrying over to the KPZ equation and Airy sheet in the scaling limit.

Core claim

When last passage percolation and directed polymer models on Z^2 possess an integrable structure coming from the RSK correspondence or the geometric RSK correspondence, their basic invariances under translation and reflection combine with a decoupling property to produce a rich collection of additional symmetries, including shift and rearrangement invariance statements for last passage times, geodesic locations, disjointness probabilities, polymer partition functions, and quenched polymer measures; the same framework also yields scrambled RSK correspondences and an RSK correspondence for moon polyominoes, and the results extend to the KPZ equation and the Airy sheet.

What carries the argument

The RSK correspondence (or geometric RSK correspondence) together with the decoupling property, which together extend the model's basic translation and reflection invariances into shift and rearrangement invariances.

If this is right

  • Last passage times remain unchanged in distribution under permitted shifts and rearrangements of the underlying weights.
  • Geodesic locations and disjointness probabilities are invariant under the same operations.
  • Polymer partition functions and quenched polymer measures inherit corresponding shift and rearrangement invariances.
  • The symmetries persist in the scaling limits given by the KPZ equation and the Airy sheet.
  • Scrambled versions of the RSK correspondence exist and an RSK correspondence can be defined for moon polyominoes.

Where Pith is reading between the lines

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

  • The rearrangement invariances could be used to reduce certain path-counting or probability calculations to equivalent problems on simpler weight configurations.
  • The same combination of basic invariances and decoupling may apply to other integrable lattice models that admit RSK-type bijections, such as certain tiling or matching problems.
  • In the continuum, the hidden invariances might translate into previously unrecognized symmetries of the directed landscape or the KPZ fixed point that are not visible from the usual translation and reflection properties alone.

Load-bearing premise

The models must possess an integrable structure from the RSK or geometric RSK correspondence and must satisfy a decoupling property.

What would settle it

Compute the distribution of last passage times in geometric last passage percolation before and after a specific weight rearrangement permitted by the claimed invariance and check whether the distributions match exactly.

Figures

Figures reproduced from arXiv: 2002.09459 by Duncan Dauvergne.

Figure 1
Figure 1. Figure 1: When M is an environment of geometric or exponential random variables, we have (ZM(u),ZM(v)) d = (ZM(u),ZM(w)). (I) For c ∈ Z2 , define (TcM)x = Mx+c. Then ZM d = ZTcM . (II) Define (R1M)(x,y) = M(y,x) . Then ZM d = ZR1M. (III) Define (R2M)(x,y) = M(−x,−y) . Then ZM d = ZR2M . For general weights, we shouldn’t necessarily expect that there are distributional symmetries beyond the ones listed above. On the … view at source ↗
Figure 2
Figure 2. Figure 2: In (a), three points u,v,w ∈ Z 4 ↑ are represented both as boxes in the plane and by sample paths from u − to u + , v − to v + and w − to w + . The pair (u,v) crosses hori￾zontally and the pair (v, u) crosses vertically. The boxes v and w are disjoint. The map f ∶ {u,v,w} → {f(u),f(v),f(w)} preserves horizontal and vertical crossings, but does not preserve disjointness, since f(u) and f(w) are disjoint but… view at source ↗
Figure 3
Figure 3. Figure 3: An example of Theorem 1.2. The distribution of (ZM(u1),ZM(u2),ZM(u3)) in (a) is the same as the distribution of (ZM(Tc1 u1),ZM(Tc2 u2),ZM(u3)) in (b). Using the language of Theorem 1.2, we can take Ui = {ui} for i = 1,2,3. THEOREM 1.2. Let M be an environment of i.i.d. exponential or geometric random vari￾ables. Let U1,... ,Uk be subsets of Z 4 ↑ such that for any i ∈ {1,... , k −1} and any ui ∈ Ui and ui+… view at source ↗
Figure 4
Figure 4. Figure 4: An example of Theorem 1.3. In the language of that theorem, we take Ui = {ui}, Vi = {vi} and f(ui) = u˜i ,f(vi) = v˜i for all i. At the level of individ￾ual boxes, f is a translation. Moreover, f preserves the disjointness of pairs ui , uj and vi ,vj for i ≠ j, and the fact that every pair (ui ,vj) crosses horizontally. There￾fore f preserves last passage values: (ZM(u1),ZM(u2),ZM(v1),ZM(v2),ZM(v3)) d = (Z… view at source ↗
Figure 5
Figure 5. Figure 5: An example of Theorem 1.4. In that theorem, we take Ui = {ui},Wi = {wi} and c = (0,−1). From the picture, we can easily check the crossing and dis￾jointness conditions of that theorem, and so (ZM(u1),ZM(u2),ZM (w1),ZM(w2)) d = (ZM(Tcu1),ZM(Tcu2),ZM(w1),ZM(w2)). THEOREM 1.4. Let M be an environment of i.i.d. exponential or geometric random vari￾ables. Let U,W ⊂ Z 4 ↑ . Suppose that we can partition U = U1 ∪… view at source ↗
Figure 6
Figure 6. Figure 6: An example of Corollary 1.7. To apply that theorem, we set U = {u} and V = {v}. The intersection u ∩ v can be taken as the box x, and the box w can be taken as the smallest box containing both u ∩ v and u ∩ Tcv; this is the shaded region. The joint distribution of the portion of the paths πM(u) and πM(v) that lie outside of the shaded region w in (a) is the same as the joint distribution of the portion of … view at source ↗
Figure 7
Figure 7. Figure 7: The Young diagram of a partition and a moon polyomin [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Corollary 1.12 is illustrated in (a). The joint distribution of the Airy sheet S on S1 and S2 is the same as the joint distribution of S on TrS1 and S2. Corollary 1.13 is il￾lustrated in (b). After an appropriate parametrization m of Γ(g), the process S ○ m has the same distribution as the two axial parabolic Airy processes S(0,⋅) and S(⋅,0). The correct parametrization is one that preserves L 1 -distance.… view at source ↗
Figure 9
Figure 9. Figure 9: An illustration of the kind of paths allowed in the d [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An example of the the map Φ from Theorem 3.3 on a 3 × 3 box for the (+,×)- algebra. the equation ZM(u j i ) = ZΦ(M)(P(u j i )) only involves entries Φ(M)x with 1 ≤ x1 ≤ i and m − j + 1 ≤ x2 ≤ m, it is straightforward to see that these equations determine Φ(M). We now first complete the proof in the case where (R,⊗,⊕) is the algebra of multiplication and addition over R>0. Construct two n × n matrices L an… view at source ↗
read the original abstract

Last passage percolation and directed polymer models on $\mathbb Z^2$ are invariant under translation and certain reflections. When these models have an integrable structure coming from either the RSK correspondence or the geometric RSK correspondence (e.g. geometric last passage percolation or the log-gamma polymer), we show that these basic invariances can be combined with a decoupling property to yield a rich new set of symmetries. Among other results, we prove shift and rearrangement invariance statements for last passage times, geodesic locations, disjointness probabilities, polymer partition functions, and quenched polymer measures. We also use our framework to find `scrambled' versions of the classical RSK correspondence, and to find an RSK correspondence for moon polyominoes. The results extend to limiting models, including the KPZ equation and the Airy sheet.

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

0 major / 3 minor

Summary. The manuscript shows that last passage percolation and directed polymer models on Z^2 possessing integrable structure from the RSK or geometric RSK correspondence (e.g., geometric LPP or log-gamma polymer) combine their basic translation and reflection invariances with a decoupling property to produce new shift and rearrangement symmetries. These apply to last passage times, geodesic locations, disjointness probabilities, polymer partition functions, and quenched measures. The framework also yields scrambled RSK correspondences, an RSK correspondence for moon polyominoes, and extensions to the KPZ equation and Airy sheet limits.

Significance. If the decoupling property is established as stated, the results supply a systematic way to generate new invariances from standard RSK properties, strengthening the toolkit for exact computations and limit theorems in integrable polymer models. The extension to KPZ/Airy objects and the construction of scrambled correspondences are notable strengths.

minor comments (3)
  1. §1: The decoupling property is introduced as an input; a self-contained statement of its precise hypotheses (including the class of models to which it applies) would improve readability before the main theorems.
  2. §3, Definition 3.2: Notation for the scrambled RSK map could be clarified by explicitly contrasting it with the classical RSK correspondence in a short table or diagram.
  3. §5: The extension to the Airy sheet is stated at a high level; adding a brief remark on how the decoupling passes to the limit would help readers track the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, as well as the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

Derivation self-contained using external RSK and decoupling inputs

full rationale

The paper takes standard translation/reflection invariances and a decoupling property as given inputs for models with RSK or geometric RSK structure, then derives new shift/rearrangement symmetries for last passage times, geodesics, partition functions, and measures. No step reduces a claimed result to a fitted parameter, self-definition, or self-citation chain; the decoupling property is an independent assumption whose consequences are derived rather than presupposed. The framework is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of integrable structure via RSK correspondences and a decoupling property in the models; these are domain assumptions standard in integrable probability but not derived in the abstract.

axioms (1)
  • domain assumption Models with RSK or geometric RSK structure possess a decoupling property that can be combined with translation and reflection invariances
    Invoked directly in the abstract to produce the new set of symmetries for LPP, polymers, and their limits.

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