Spectral chaos in discrete classical systems

Spectral chaos - the emergence of universal spectral correlations described by random matrix theory - is widely used as a diagnostic of quantum chaos. In this seminar, I will discuss how this notion in the context of classical systems. I will focus on reversible discrete dynamical systems, where spectral correlations are directly related to the statistics of periodic orbits. I will first introduce ensembles of random permutations as a counterpart to random matrix ensembles and discuss their orbit statistics. I will then examine how and why similar statistics can emerge in permutation circuits. In short, I will try to convince you that spectral chaos is a notion applicable to both classical and quantum systems.