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arxiv: 2405.00194 · v4 · pith:IWZXJZGRnew · submitted 2024-04-30 · 🧮 math.PR · math-ph· math.CO· math.MP

The directed landscape from Brownian motion

Duncan Dauvergne , B\'alint Vir\'ag This is my paper

Pith reviewed 2026-05-24 01:12 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.COmath.MP
keywords directed landscapeBrownian motionRSK correspondencestationary horizonlast-passage percolationAiry line ensemblePitman operatorsbiHecke monoid
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The pith

An almost sure bijection recovers the directed landscape on the half-plane from a sequence of independent Brownian motions as the scaling limit of RSK.

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

The paper constructs a map turning sequences of independent Brownian motions into the directed landscape with probability one. This bijection is identified as the continuous limit of the combinatorial Robinson-Schensted-Knuth correspondence. The construction supplies an explicit inverse and a coupling making Brownian last-passage percolation converge in probability to the landscape. Readers would care because it supplies a concrete generative description of the directed landscape from elementary processes and proves that the landscape on a strip can be recovered from the parabolic Airy line ensemble.

Core claim

We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson-Schensted-Knuth (RSK) correspondence. The Brownian motions arise as the marginals of the multi-path stationary horizon associated with the directed landscape. The inverse map is fully explicit and yields a natural coupling in which Brownian last-passage percolation converges in probability to the directed landscape. As an application, we prove that the directed landscape restricted to a strip can be reconstructed from the parabolic Airy line ensemble.

What carries the argument

The almost sure bijection from sequences of independent Brownian motions to the directed landscape, realized as the scaling limit of the RSK correspondence.

Load-bearing premise

The input Brownian motions must coincide exactly with the marginals of the multi-path stationary horizon associated with the directed landscape.

What would settle it

A positive-probability failure of the inverse map to recover the original directed landscape when started from the marginals of its own stationary horizon.

Figures

Figures reproduced from arXiv: 2405.00194 by B\'alint Vir\'ag, Duncan Dauvergne.

Figure 1
Figure 1. Figure 1: A disjoint optimizer for k = 3 from ((0, 0, 0.2), 5) to ((0.7, 0.9, 1), 1). Note that in our coordinate system, we view paths as moving up and to the right across the page. for multi-path last passage values in the situations that occur most, we write x k ∶= (x, . . . , x) ∈ R k . Typically, we will consider multi-path last passage when all the points in p lie on a common vertical or horizontal line (and s… view at source ↗
Figure 2
Figure 2. Figure 2: Semi-infinite geodesics in B, indexed by the direction θ ∈ R. To make sense of the direction when θ < 0, we look along down-right paths, or equivalently, paths in the reversed environment. Theorem 1.6 (Theorem 1.5, [DOV22]). For every a ∈ R, let B a ∈ C Z (R) be an environment of independent two-sided Brownian motions of common drift a. Let (x, s)a = (x − as/4,⌊sa3 /8⌋ + 1). Then as a → −∞, L a (x, s; y, t… view at source ↗
Figure 3
Figure 3. Figure 3: Semi-infinite geodesics in L. The colouring of paths in Figures 2 and 3 is chosen so that the start points and directions match in the two pictures under the version of the RSK correspondence in Theorem 1.7. where the maximum is over all disjoint k-tuples of paths π = (π1, . . . , πk) with π(r) ∈ R k < for r ∈ (s, t), and such that π(s) = x, π(t) = y. In [DZ21], the authors show that the maximum in (14) is… view at source ↗
read the original abstract

We construct an almost sure bijection that recovers the directed landscape on the half-plane from a sequence of independent Brownian motions. This map is the natural scaling limit of the Robinson--Schensted--Knuth (RSK) correspondence. The Brownian motions arise as the marginals of the multi-path stationary horizon associated with the directed landscape. The inverse map is fully explicit and yields a natural coupling in which Brownian last-passage percolation converges in probability to the directed landscape. As an application, we prove that the directed landscape restricted to a strip can be reconstructed from the parabolic Airy line ensemble, resolving a conjecture of the first author and Zhang. Along the way we develop two new versions of RSK in the semi-discrete setting, introduce a general theory of sorting via Pitman operators that generates a faithful action of the biHecke monoid, and establish key identities for the multi-path stationary horizon for both the directed landscape and Brownian last-passage percolation.

Editorial analysis

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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

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Summary. The paper constructs an almost sure bijection recovering the directed landscape on the half-plane from a sequence of independent Brownian motions, realized as the scaling limit of a new semi-discrete RSK correspondence. The Brownian motions are identified as marginals of the multi-path stationary horizon; the inverse map is explicit and yields a coupling in which Brownian LPP converges in probability to the directed landscape. Along the way the authors introduce two new semi-discrete RSK variants, develop a general Pitman-operator theory generating a faithful biHecke monoid action, establish key identities for the multi-path stationary horizon in both the directed landscape and Brownian LPP settings, and apply the construction to reconstruct the directed landscape restricted to a strip from the parabolic Airy line ensemble, thereby resolving a conjecture of the first author and Zhang.

Significance. If the central bijection and the supporting identities hold, the work supplies an explicit, parameter-free coupling between the directed landscape and Brownian motions together with a fully constructive inverse; this is a substantial contribution to the theory of the directed landscape and last-passage percolation. The resolution of the Airy-line-ensemble conjecture is a concrete advance, and the introduction of the biHecke monoid action via Pitman operators provides new algebraic machinery whose utility may extend beyond the present setting. The manuscript ships machine-checked-style identities for the stationary horizon and an explicit inverse, both of which strengthen the result.

minor comments (3)
  1. The definition of the multi-path stationary horizon is invoked in the abstract and introduction before its precise construction appears; a forward reference or one-sentence reminder in §1 would improve readability for readers unfamiliar with the earlier literature.
  2. Notation for the biHecke monoid generators and the Pitman operators is introduced gradually; collecting the main relations in a single displayed block early in the Pitman-operator section would aid cross-referencing.
  3. Several statements of convergence in probability are stated without an explicit rate; while not required for the main claims, a brief remark on the mode of convergence would clarify the strength of the coupling result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions detected

full rationale

The central result is an explicit almost-sure bijection constructed as the scaling limit of newly developed semi-discrete RSK maps together with an internal theory of Pitman operators that generates the biHecke action; the inverse is given explicitly and the required marginal identifications for the multi-path stationary horizon are established directly inside the manuscript for both the directed landscape and Brownian LPP. The resolution of the earlier conjecture appears only as a downstream application and is not used to justify any step of the bijection. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on the established definitions and properties of Brownian motion, the directed landscape, and the stationary horizon from prior literature.

axioms (1)
  • domain assumption Properties of the directed landscape and its multi-path stationary horizon as previously defined
    The bijection is constructed on top of these objects.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Invariant measures for half-space geometric LPP: classification and the one force--one solution principle

    math.PR 2026-05 unverdicted novelty 8.0

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

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    Almost surely, B0 = B

    work page

  2. [2]

    For any a ∈R, Ba is a sequence of independent Brownian motions with drift a. 52

    work page

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    Suppose also that λ + a ∶= ( λ 1 + a,

    Fix a ∈R and λ ∈( −∞, 0] k ≤ ∪[ 0, ∞) k ≤. Suppose also that λ + a ∶= ( λ 1 + a, . . . , λ k + a) ∈( −∞, 0] k ≤ ∪[ 0, ∞) k ≤. Then a.s. for all x ∈Rk ≤ we have that W λ + a( x; B) = W λ ( x; Ba) . (54)

    work page

  4. [4]

    For any a, b ∈R, almost surely we have that: ( Ba) −a = B, ( Ba) b = Ba+ b. (55) In particular, if for a ∈R we define a ∶CN( R) → C N( R) by letting a( f ) = f a( x) −ax, then for any countable subgroup G ⊂R, this defines a measure- preserving G-action on a µ N-almost sure subset of CN( R) . Remark 4.13. Theorem 4.12 allows us to naturally extend the definit...

    work page

  5. [5]

    (57) Now, by Theorem 4.9 we have that ( W m) 0 d= ( ˜W m) 0 ∼µ θm

    for λ and x ∈Rℓ ≤ we have that W λ n ( x; Ba) = ˜W m[( −∞, n + m) →( x, 1)] . (57) Now, by Theorem 4.9 we have that ( W m) 0 d= ( ˜W m) 0 ∼µ θm. Moreover, W m i ( 0) = ˜W m i ( 0) = 0 for i ≤m and we can extract the values of W m i ( 0) , i ≥m + 1 from ( W m) 0 by the formula in Theorem 4.9.3. The same formula extracts the values of ˜W m i ( 0) , i ≥m + 1...

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    Case 2: a + λ ∈[ 0, ∞) k ≤, a > 0, λ ∈( −∞, 0] k ≤

    as m →∞ to get that ( W λ + a( x; B) , W λ ( x; Ba)) d= ( W λ ( x; Ba) , W λ ( x; Ba)) , 54 which implies that W λ + a( x; B) = W λ ( x; Ba) almost surely, as desired. Case 2: a + λ ∈[ 0, ∞) k ≤, a > 0, λ ∈( −∞, 0] k ≤. In this case, let θm = ( λ + a, a m) ∈ [ 0, ∞) m+ k ≤ , and form W m from θm as above. Let PRSK be the Pitman transform introduced prior ...

    work page

  7. [7]

    Call these geodesics π θ,p and π θ,p R

    (Theorem 3.14, [ R V21]) For every point p = ( x, s ) ∈R2 and every direction θ ∈R, there exist leftmost and rightmost semi-infinite geodesic s ending at p in direction θ. Call these geodesics π θ,p and π θ,p R . For fixed p, θ , almost surely π θ,p = π θ,p R

    work page

  8. [8]

    (Theorem 6.3(i), [ BSS24]) For any t < s, the functions ( θ, s ) ↦π θ, ( x,s ) ( t) and ( θ, s ) ↦π θ, ( x,s ) R ( t) are nondecreasing in θ, s

    work page

  9. [9]

    (Theorem 2.5, [ BSS24]) There exists a random, translation invariant, count- ably infinite, dense subset Ξ ⊂R such that for θ ∉Ξ, if π, π ′are two semi- infinite geodesics in direction θ, then π ( s) = π ′( s) for all small enough s

    work page

  10. [10]

    Then the Busemann function Bθ( p; L) ∶= lim s→−∞ L( ¯π ( s) ; p) −L( ¯π ( s) ; 0, 0) 56 exists and does not depend on the choice of π

    (Theorem 5.1, [ BSS24]) Consider θ ∉Ξ, p = ( y, t ) ∈R2, and any path π ∶ ( −∞, t ] →R with π ( s)/slash.left/divides.alt0 s/divides.alt0 →θ as s →∞. Then the Busemann function Bθ( p; L) ∶= lim s→−∞ L( ¯π ( s) ; p) −L( ¯π ( s) ; 0, 0) 56 exists and does not depend on the choice of π . Here recall the notation ¯π ( s) ∶= ( π ( s) , s ) . Moreover, for fixed...

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  11. [11]

    We can now state analogues of the main results of Section 4.1 for the directed landscape

    (Theorem 4.3, [ R V21]) For s < t and y ∈R, we have the metric composition law Bθ( y, t ; L) = max x∈R Bθ( x, s ; L) + L( x, s ; y, t ) . We can now state analogues of the main results of Section 4.1 for the directed landscape. The proofs are almost identical to the proofs for Brownian LPP, so we will go through the steps briefly. For x ∈Rk ≤ and θ ∈J k < ...

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  12. [12]

    , π θ, ( yk,t ) ) and π θ, z ∶= ( π θ, ( z1,t ) ,

    For every k ∈N, a > 0, t ∈R and every compact box K = ∏k i= 1[ θ− i , θ + i ] ⊂Rk < there exist vectors y, z ∈Qk ≤ with y ≤−ak ≤ak ≤z such that for any θ ∈K the k-tuples of semi-infinite leftmost geodesics π θ, ( y,t ) ∶= ( π θ, ( y1,t ) , . . . , π θ, ( yk,t ) ) and π θ, z ∶= ( π θ, ( z1,t ) , . . . , π θ, ( zk,t ) ) are both disjoint k-tuples. In particu...

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  13. [13]

    , k , any geodesics in direction θ1,i to ( yi, t ) and direction θ2,i to ( zi, t ) are equal at time T

    For any a > 0, t ∈R, θ ∈J k < , there exists ǫ > 0, there exists y ≤−ak < ak ≤z and T < t such that whenever /parallel.alt1 θ1 −θ/parallel.alt1 ∞ < ǫ, /parallel.alt1 θ2 −θ/parallel.alt1 ∞ < ǫ: • For any i = 1, . . . , k , any geodesics in direction θ1,i to ( yi, t ) and direction θ2,i to ( zi, t ) are equal at time T . • Any geodesics from θ1,i to ( yi, t...

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  14. [14]

    , π k) with direction θ

    exists for all k ∈N, θ ∈J k < , x ∈Rk ≤, t ∈R, and equals lim s→−∞ L( ¯π ( s) ; x, t ) − k /summation.disp j= 1 L( ¯π ( s) ; 0, 0) for any k-tuple of paths π = ( π1, . . . , π k) with direction θ

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  15. [15]

    There exist semi-infinite optimizers in every direction θ ∈Rk ≤ ending at every point in ( x, s ) ∈Rk ≤ ×R. 57

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    For any θ ≤θ′, x ≤x′and t ∈R we have Bθ( x′, t ) + Bθ′ ( x, t ) ≤Bθ( x, t ) + Bθ′ ( x′, t )

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  17. [17]

    If /parallel.alt1 θ −θ′/parallel.alt1 ∞ < ǫ and θ′∈J k < as well, then for all x ∈[ −a, a ] k ≤, Bθ( x, t ) −Bθ( 0k, 0) = Bθ′ ( x, t ) −Bθ′ ( 0k, 0)

    Let θ ∈J k < , a ∈R, ǫ > 0 be as in part 2 above. If /parallel.alt1 θ −θ′/parallel.alt1 ∞ < ǫ and θ′∈J k < as well, then for all x ∈[ −a, a ] k ≤, Bθ( x, t ) −Bθ( 0k, 0) = Bθ′ ( x, t ) −Bθ′ ( 0k, 0)

    work page

  18. [18]

    For x, θ ∈Rk ≤ and t ∈R define ˆBθ( x, t ) = lim θ′↓θ,θ ′∈I k< Bθ′ ( x, t ) −Bθ′ ( 0k) (62) Bθ( x, t ) = lim θ′↓θ,θ ′∈I k< Bθ′ ( x, t ) − m( θ) /summation.disp j= 1 Bθ′Ij ( θ ) ( 0/divides.alt0 Ij ( θ)/divides.alt0 ) (63) Then the limits above exists for all fixed x ∈Rk ≤, t ∈R. Proof. Part 1 follows as in the proof of Lemma 4.4, uses that translation invar...

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  19. [19]

    We first deal with the case when F ⊂⋃∞ k= 1 Rk <

    when ( θ, x ) is restricted to Kb and convergence of the first coordinates is with respect to the L∞-norm on functions from Kb → R ∪{ ±∞} . We first deal with the case when F ⊂⋃∞ k= 1 Rk <. We will also assume that F has the following downward closed property: if θ ∈F ∩Rk < and I ⊂{ 1, . . . , k } , then θI ∈F as well. We appeal to the framework of Lemma 5....

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  20. [20]

    Since π is an optimizer we also have that Bθ r( x, t ; L) = Bθ r( π ( s) , s ; L) + L( π ( s) , s ; y, t )

    holds with ¯Bθ r in place of ¯Bθ on both sides by the usual metric composition law. Since π is an optimizer we also have that Bθ r( x, t ; L) = Bθ r( π ( s) , s ; L) + L( π ( s) , s ; y, t ) . for all r < s. Combining these facts with the pointwise convergence Bθ r → ¯Bθ = Bθ from Corollary 5.5 yields the result. Lemma 5.7. Fix θ ∈Rk ≤, x ∈Rk ≤, and t ∈R....

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  21. [21]

    (Law of the RSK image) The joint law of ( Ba, a ∈R) is same as the joint law of the processes ( Ba, a ∈R) defined in Theorem 4.12. In particular, each Ba is a sequence of independent two-sided Brownian motions of dri ft a and we have the Busemann isometry W θ( x; Ba) = B( θ+ a)/slash.left 2( x; L) for all x, θ ∈Rk ≤, k ∈N

    work page

  22. [22]

    There is a measurable function f ∶CN( R) → C( X− ↑) such that almost surely f ( B0) = L/divides.alt0 X− ↑

    (Abstract Invertibility) Define H2 ↑ ∶= {( x, s ; y, t ) ∈R4 ↑ ∶t ≤0} . There is a measurable function f ∶CN( R) → C( X− ↑) such that almost surely f ( B0) = L/divides.alt0 X− ↑

    work page

  23. [23]

    (Explicit Inversion Formula 1) For every a ∈R, as in Theorem 2.11, for ( x, s ; y, t ) ∈X− ↑∶= {( x, s ; y, t ) ∈X↑∶t ≤0} define La( x, s ; y, t ) = Ba[( x, s ) a →( y, t ) a] −a2 4 ( t −s) , where ( x, s ) a = ( x −as/slash.left 4, /uni2308.alt1 sa3/slash.left 8/uni2309.alt1 + 1) . Then as a →−∞, La converges to L in probability, in either of the followin...

    work page

  24. [24]

    Then as a →−∞, La converges to L in probability as functions in the compact topology on X↑

    (Explicit Inversion Formula 2) For every a ∈R, extend the Brownian line environments Ba to environments indexed by i ∈Z by the rule that: k /summation.disp i= 1 Ba 1−i( x) = lim t→∞ L( 0k, 0; −( a/slash.left 2) kt, t ) −L( xk, 0; −( a/slash.left 2) k t, t ) , and use this to extend the definition of La to all of X↑. Then as a →−∞, La converges to L in prob...

    work page

  25. [25]

    As n →∞, ( Xn, Y n) d →( X, Y ) and Xn P →X

    work page

  26. [26]

    Then Yn P →Y as n →∞

    There is a measurable function h ∶S →T such that Y = h( X) a.s. Then Yn P →Y as n →∞. Proof. For every ǫ > 0, by Lusin’s theorem we can find a closed set E ⊂S such that h/divides.alt0 E is continuous and P( X ∉E) ≤ǫ. Next, Dugundji’s extension theorem [ Dug51] states that if S is any metric space, E ⊂S is closed, and T is a locally convex topological vecto...

    work page

  27. [27]

    Pick θ′∈ Z ∩( θ, ∞)

    as the proof of ( 75) is similar but simpler. Pick θ′∈ Z ∩( θ, ∞) . For every n ∈Z, let π n denote the leftmost semi-infinite geodesic to ( n, 0) in direction θ′. By translation invariance of the directed landscape, ther e exists some random N ∈Z such that if n ≥N then π n( s) > x. Therefore for n > N we have: Bθ( w /divides.alt0 Su) = lim r→−∞ max z1≥x,z ...

    work page

  28. [28]

    Fix x ∈{ x1, x 2} , y ∈{ y1, y 2}

    in fact holds almost surely. Fix x ∈{ x1, x 2} , y ∈{ y1, y 2} . By Lemma 5.14 we have that B−a( a /divides.alt0 S{ x,y } ) = max z1≥0,z 2≤0 B−a( z1 + x, s ) + L{ z1 + x, z 2 + y} + L( z2 + y, t ; a, 0) . (78) We aim to show that B−a( a /divides.alt0 S{ x,y } ) = B−a( x, s ) + L{ x, y } + L( y, t ; a, 0) + Y a( x, y ) , (79) 72 where the error Y a( x, y )...

    work page

  29. [29]

    the geodesic πλ,w is disjoint from the path π n n

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  30. [30]

    F n( λ, w ) = Bλ ( w) . Proof. This follows since B(( λ, ( n1/slash.left 3) n)) ( w, ( n1/slash.left 3/slash.left 2) n) = B( n1/slash.left 3) n (( n1/slash.left 3/slash.left 2) n) + Bλ ( w) if and only if there is an optimizer in direction ( n1/slash.left 3) n to (( n1/slash.left 3/slash.left 2) n, 0) which is disjoint from a geodesic to ( w, 0) in direct...

    work page arXiv 2024