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The KPZ equation and the directed landscape
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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.
Forward citations
Cited by 9 Pith papers
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Invariant measures and shocks in the KPZ fixed point
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...
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Spatial decorrelation of KPZ from narrow wedge
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.
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Exceptional force, uncountably many solutions in the KPZ fixed point
For exceptional slopes, the KPZ fixed point has uncountably many eternal solutions, classified by bi-infinite mixed Busemann interfaces.
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An upper tail field of the KPZ fixed point
A newly constructed 'upper tail field' is the local limit of the KPZ fixed point near a conditioned large value, interpolating between Brownian and KPZ scaling regimes.
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Applications of optimal transport to Dyson Brownian Motions and beyond
A new optimal transport argument shows that Dyson Brownian motions with beta at least 2, and their scaling limits, have Brownian-like uniform modulus of continuity bounds with constants independent of the particle layer.
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Periodic KPZ fixed point with general initial conditions
For general periodic initial data, the relaxation-time-scale limit of periodic TASEP defines the periodic KPZ fixed point with explicit multipoint distributions.
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The half-space KPZ line ensemble and its scaling limit
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...
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Two-time spatial decorrelation for the flat KPZ fixed point
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.
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Dynamic scaling of growing interfaces
An expert review of the KPZ equation's history, mathematical developments, and applications, with no new research results.
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