Random Matrix Theory. Given a problem with a large number of degrees of freedom or some amount of disorder, we are unable to give an exact description of the dynamics and must instead rely on a statistical description. An example of such a problem might be the decay of heavy nuclei or scattering in a disordered cavity. For this we need a model that is both rich enough to capture the essential features of the underlying system and simple enough to admit an exact solution. The question that one is then led to ask is: what properties of the system are relevant for the dynamics? One might expect that, analogous to the theory of phase transitions in statistical mechanics, many of the microscopic details have no bearing on the qualitative behaviour. If we consider a sufficiently complex quantum system of dimension N , these microscopic details are encoded by either the Hamiltonian H or the time evolution operator U = e−iHt, and our task is to replace H or U by an appropriate random N × N matrix. When N is large, we hope that a kind of central limit theorem will apply to allow us to conclude that the spectral properties of a generic Hamiltonian are close to those of the random Hamiltonian. If the system has a set of symmetries described by a group G, then, according to a classification in [1], the set of matrices invariant under G is a direct product of three irreducible components labelled by β and corresponding to the presence of both time reversal symmetry and rotational invariance (β = 1), absence of time reversal symmetry (β = 2) and the presence of time reversal symmetry and half-integer angular momentum (β = 4). The invariance requirement leads to a duality requirement on the set of Hermitian or unitary matrices, summarised in the table below β H U 2 Hermitian unitary 4 quaternion real self-dual quaternion The randomly chosen matrix must at the very least be compatible with the symmetry of the system. We are then left to determine a probability measure on the corresponding matrix space with Lebesgue measure Y dH = cN,β dHij, ij where the product is taken over independent matrix elements. Assuming that there is no preferred basis, the matrix elements with respect to any two bases should be distributed according to the same law. This leads to the invariance of the measure under orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) transformations. With the additional assumption that the matrix elements be statistically independent (except for the dependence imposed by symme...
