Chaotic processes, erratic functions and random matrix theory

by Jacob Sonnenschein

Wednesday 22 Apr 2026, 15:10 → 16:10 Europe/Rome

GGI - Room B

Chaotic processes often admit a description in terms of erratic functions of a certain continuous variable. Examples of such a behavior are the scattering angle as a function of the incident angle in a pinball experiment, the leaky torus phase shift as a function of the wave-number and the decay amplitude of highly excited string state (HES) into two low-mass states or a scattering amplitude of a HES with three low-mass states as a function of an angle. I will present a novel measure of chaotic behavior based on a map between the set of maxima of the erratic functions and the eigenvalues of random matrices and the corresponding spectral form factor. I will introduce the notion of multi-dimensional chaos that applies to processes described by erratic functions of several dynamical variables.The classical and quantum scattering off a pinball system will be described. I will show that the eigenvalues of the S-matrix are distributed according to the Circular Orthogonal Ensemble (COE) in random matrix theory (RMT), provided the setup be asymmetric and the wave-number be large enough. I will then consider the electric potential associated with charges randomly located on a plane as a toy model that generalizes the scattering from a leaky torus. I will propose several methods to analyze the two dimensional spacings between the extrema of this function. The latter follow a repulsive Gaussian β-ensemble distribution even for Poisson-distributed positions of the charges. These methods will be applied to the case of a chaotic S-matrix and of the quantum pinball scattering. The spacings between nearest neighbor extrema points and ratios between adjacent spacings for this case follow a logistic and Beta distributions correspondingly. A generalization of the spectral form factor will be introduced and determined. We conjecture about a potential relation with random tensor theory.