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.

Disclosure

“59981) from the Simons Foundation. F.P. was supported in part by National Natural Science Foundation of China (No. 12571153) and National Key R&D Program of China (No. 2022YFA 1006500). During the preparation of this work, the authors used large language model–based tools as auxil- iary aids in the research and writing process. All mathematical content was reviewed and verified by the authors, who take full responsibility for the manuscript. References”

PDF page 22

Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-23T18:58:33.486883Z

Structural counts