平坦 KPZ 不动点的二次空间去相关
Two-time spatial decorrelation for the flat KPZ fixed point
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中文总结 AI 辅助
研究平坦 KPZ 不动点的二次空间去相关问题，通过结合 Airy₁ 过程的三次指数混合与定位估计进行证明，得出经\(N^{1/2}\)归一化的中心空间平均值收敛到特定高斯过程的结果。
中文摘要 AI 辅助
我们建立了具有平坦初始数据的 Kardar-Parisi-Zhang 不动点的定量二次空间去相关。对于任意\(s,t>0\)，存在常数\(C,c>0\)使得当\(|x|\geq1\)时，\(\big|{\rm Cov}(\mathfrak{h}(t,x),\mathfrak{h}(s,0))\big| \leq C\exp\{-c|x|^3\}\)。与直接由 Airy₁ 过程控制的固定时间协方差不同，二次协方差涉及整个早期高度轮廓的非线性变分演化。我们的证明将 Airy₁ 过程的三次指数混合与有向景观中中间优化器的均匀定位估计相结合。结果，通过\(N^{1/2}\)归一化的中心空间平均值在有限维分布中收敛到一个中心高斯过程，其协方差是平坦 KPZ 不动点的空间积分二次相关性。
英文摘要
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.
