For fixed t>0, Cov[h(t,x),h(t,0)] ∼ t/x as x→∞, and the spatial average over [0,N] normalized by sqrt(N log N) converges to sqrt(2) times Brownian motion.
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Spatial decorrelation of KPZ from narrow wedge
For fixed t>0, Cov[h(t,x),h(t,0)] ∼ t/x as x→∞, and the spatial average over [0,N] normalized by sqrt(N log N) converges to sqrt(2) times Brownian motion.