Two-time spatial decorrelation for the flat KPZ fixed point

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 \[ \big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \le C\exp\{-c|x|^3\},\qquad |x|\ge1. \] Unlike the fixed-time covariance, which is governed directly by the Airy${}_1$ process, the two-time covariance involves the nonlinear variational evolution of the entire earlier height profile. Our proof combines cubic-exponential mixing of the Airy${}_1$ 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.