Stationary $1/f^{\alpha}$ noise in discrete models of the Kardar-Parisi-Zhang class

TL;DR

This study investigates stationary 1/f^α noise in discrete models of the KPZ class, revealing non-exponential correlations, finite-size effects, and a spectral exponent of 5/3, with implications for understanding interface fluctuations.

Contribution

It demonstrates the existence of stationary 1/f^α noise in discrete KPZ models and characterizes the spectral and correlation properties with finite-size scaling analysis.

Findings

Correlation function is non-exponential and vanishes at a diverging time.

Power spectra exhibit 1/f^α scaling with α=5/3.

Finite-size scaling confirms dynamic scaling of correlations.

Abstract

In discrete models describing growing rough interfaces of the Kardar-Parisi-Zhang universality class, we examine height fluctuations at a fixed site as a function of time in the monolayer unit. For small systems, we show that it is possible to reach the stationary state. We compute the two-time autocorrelation and power spectra independently. The correlation function remains non-exponential and vanishes after a correlation time that diverges with system size. As a result, the power spectra display a lower cutoff that maintains constant power. In the nontrivial frequency regime, we observe $1/f^{\alpha}$-type scaling with the spectral exponent 5/3. Finite-size scaling reveals that the temporal correlation function follows a dynamic scaling. Our findings, supported by scaling-theoretical arguments, establish that the fluctuations are wide-sense stationary, implying applicability of the Wiener-Khinchin theorem.