The KPZ equation describes interface growth — a surface driven by noise, smoothed by diffusion, steepened by nonlinearity. The one-point statistics (how high is the surface at a given location) are well-understood through exact solutions. The two-point spatial statistics (how correlated are two locations at the same time) have been much harder.
Chen and Jiménez determine the spatial covariance for flat initial data — the surface starts level. Using Malliavin calculus and the Clark-Ocone representation, they prove the covariance decays as a Gaussian at large spatial separation: exponential suppression with a decay rate proportional to |x|². This is fast decay. Points far apart on a flat-initialized KPZ surface are nearly independent.
The contrast with narrow-wedge initial data is sharp. When the surface starts from a point (narrow wedge), the covariance decays as |x|⁻¹ — polynomial, not Gaussian. The initial condition determines not just the transient behavior but the fundamental character of spatial correlations at fixed time.
The mechanism is a boundary-layer effect near time zero. For flat initial data, the early-time dynamics average the noise over the entire initial surface, creating a smoothing effect that persists into the large-time spatial statistics. For narrow-wedge data, all information originates from a single point, and the outward propagation maintains long-range correlations.
The first exact spatial covariance asymptotic for KPZ under flat initial data. The result also gives a closed-form second moment for the continuum directed random polymer partition function — the object whose free energy is the KPZ surface height.