2026/07/19 by Le Chen, Fei Pu
#math.PR
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 |\rm Cov(\mathfrakh(t,x),\mathfrakh(s,0))| ≤ Cexp\-c|x|3\, |x|≥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 N1/2, 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.