Spectral Form Factor and Random Walks
Abstract
This work examines the relationship between the spectral form factor (SFF) and planar uniform random walks. The SFF is a measure used to identify the onset of quantum chaos and scrambling in models that capture certain aspects of black holes. We demonstrate that the moments of the SFF in generic quantum chaotic systems exhibit similar behavior to the moments of the distribution describing the positions of random walkers in the Euclidean plane. This similarity holds for the mean, variance, skewness, kurtosis, and all higher-order moments of the underlying probability distribution function. Additionally, we suggest a potential generalization of planar random walks that avoid intersections along the trajectory and explore potential applications to integrable quantum systems.
