44. chen.pu:26:two-time

Two-time spatial decorrelation for the flat KPZ fixed point

Le Chen and Fei Pu

Abstract: We establish quantitative two-time spatial decorrelation for the Kardar–Parisi–Zhang fixed point with flat initial data. For every \(s,t>0\), there exist constants \(C,c>0\) such that

\[\left|\operatorname{Cov}(h(t,x),h(s,0))\right| \leq C\exp\{-c|x|^3\}, \qquad |x|\geq 1.\]

Unlike the fixed-time covariance, which is governed directly by the Airy₁ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy₁ process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, the centered spatial averages, normalized by \(N^{1/2}\), converge in finite-dimensional distributions to a centered Gaussian process whose covariance is the space-integrated two-time correlation of the flat KPZ fixed point.

Preprint

[CP26] Le Chen & Fei Pu (2026) ‘Two-time spatial decorrelation for the flat KPZ fixed point’, preprint arXiv:2607.17113, 24 pages

@article{chen.pu:26:two-time,
   title         = {Two-time spatial decorrelation for the flat {KPZ} fixed point},
   author        = {Le Chen and Fei Pu},
   year          = {2026},
   month         = {July},
   journal       = {Preprint arXiv:2607.17113},
   url           = {http://arXiv.org/abs/2607.17113}
}

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