Growing interfaces in the Kardar-Parisi-Zhang universality class develop roughness over time — the surface height at a fixed point fluctuates as material is deposited. These fluctuations are correlated in time: the height now depends on the height a moment ago. The question is what the temporal power spectrum looks like.
This paper answers it for discrete KPZ models in the stationary regime. The power spectrum follows 1/f^α scaling with a spectral exponent of exactly 5/3. This is not 1/f noise (exponent 1) or white noise (exponent 0) or Brownian motion (exponent 2), but a specific intermediate value determined by the KPZ scaling exponents.
The correlation function is non-exponential and vanishes after a correlation time that diverges with system size. This creates a lower-frequency cutoff in the power spectrum that maintains constant power — the system has a finite memory set by its physical extent. Above this cutoff, the 5/3 scaling holds.
Crucially, the fluctuations are wide-sense stationary: the mean and autocorrelation depend only on time differences, not absolute time. This validates the use of the Wiener-Khinchin theorem — the power spectrum is genuinely the Fourier transform of the autocorrelation, not an artifact of nonstationarity. The KPZ class, famous for its anomalous spatial scaling, produces equally precise temporal scaling. The roughness has a rhythm, and the rhythm has an exact exponent.