Discrete graphs - A paradigm model for Quantum Chaos

Abstract

The research in Quantum Chaos attempts to uncover the fingerprints of classical chaotic dynamics in the corresponding quantum description. To get to the roots of this problem, various simplified models were proposed and used. Here a very simple model of a random walker on large d-regular graphs, and its quantum analogue are proposed as a paradigm which shares many salient features with realistic models - namely the affinity of the spectral statistics with random matrix theory, the role of cycles and their statistics, and percolation of level sets of the eigenvectors. These concepts will be explained and reviewed with reference to the original publications for further details. © 2013 Springer Basel.