Random-Matrix Theory

Abstract

A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group of canonical transformations (see Chap. 2). In particular, the level spacing distribution P(S) generally takes the form characteristic of the universality class defined by the canonical group. Most notable among the exceptions barred by the (mathematically hard to substantiate) term ``untypical'' are systems with ``localization'' that will be discussed in Chap. 7. Conversely, ``generic'' classically integrable systems with at least two degrees of freedom tend to display universal local fluctuations of yet another type, to be considered in Chap. 5.