Spatial decorrelation of KPZ from narrow wedge

We study the spatial decorrelation of the solution to the KPZ equation with narrow wedge initial data. For fixed $t>0$, we determine the decay rate of the spatial covariance function, showing that $\mathrm{C}\mathrm{o}\mathrm{v}[h(t,x),h(t,0)]\sim \frac{t}{x}$ as $x\to \mathrm{\infty }$. In addition, we prove that the finite-dimensional distributions of the properly rescaled spatial average of the height function converge to those of a Brownian motion.