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The KPZ equation and the directed landscape

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arxiv 2301.00547 v3 pith:5ZE75NFC submitted 2023-01-02 math.PR

Xuan Wu This is my paper
classification math.PR
keywords convergencedirectedlandscapeequationairyconditionsfixedinitial
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This paper proves the convergence of the narrow wedge solutions of the KPZ equation to the Airy sheet and the directed landscape in the locally uniform topology. This is the first convergence result to the Airy sheet and the directed landscape established for a positive temperature model. We also give an independent proof for the convergence of the KPZ equation to the KPZ fixed point for general initial conditions in the locally uniform topology. Together with the directed landscape convergence, we show the joint convergence to the KPZ fixed point for multiple initial conditions.

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Cited by 6 Pith papers

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

  1. Invariant measures and shocks in the KPZ fixed point

    math.PR 2025-08 conditional novelty 8.0 of 10

    Every extremal invariant measure of the recentered KPZ fixed point is a Brownian motion with drift, and new shock-frame measures of Brownian plus Bessel form are constructed and shown to arise from open-boundary stati...

  2. Spatial decorrelation of KPZ from narrow wedge

    math.PR 2025-06 accept novelty 8.0 of 10

    For fixed t>0, Cov[h(t,x),h(t,0)] ∼ t/x as x→∞, and the spatial average over [0,N] normalized by sqrt(N log N) converges to sqrt(2) times Brownian motion.

  3. Periodic KPZ fixed point with general initial conditions

    math.PR 2026-03 conditional novelty 7.0 of 10

    For general periodic initial data, the relaxation-time-scale limit of periodic TASEP defines the periodic KPZ fixed point with explicit multipoint distributions.

  4. The half-space KPZ line ensemble and its scaling limit

    math.PR 2025-06 conditional novelty 7.0 of 10

    The half-space KPZ line ensemble is constructed, proven tight under 1:2:3 scaling in critical and supercritical regimes, and shown to have subsequential limits with a one-sided Gibbs property, including pairwise pinne...

  5. Two-time spatial decorrelation for the flat KPZ fixed point

    math.PR 2026-07 conditional novelty 6.0 of 10

    The two-time spatial covariance of the flat KPZ fixed point decays as exp(-c|x|^3), and normalized spatial averages converge to a Gaussian process with covariance equal to the space-integrated two-time correlation.

  6. Dynamic scaling of growing interfaces

    cond-mat.dis-nn 2025-07 unverdicted novelty 1.0 of 10

    An expert review of the KPZ equation's history, mathematical developments, and applications, with no new research results.

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