  Quantum counterparts of certain simple classical systems can exhibit chaotic
behaviour through the statistics of their energy levels and the irregular
spectra of chaotic systems are modelled by eigenvalues of infinite random
matrices. We use known bounds on the distribution function for eigenvalue
spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real
symmetric matrices and show that gamma distributions, which have an important
uniqueness property, can yield an approximation to the GOE distribution. That
has the advantage that then both chaotic and non chaotic cases fit in the
information geometric framework of the manifold of gamma distributions, which
has been the subject of recent work on neighbourhoods of randomness for general
stochastic systems. Additionally, gamma distributions give approximations, to
eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random
hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite
random hermitian matrices with real quaternionic elements, except near the
origin. Gamma distributions do not precisely model the various analytic systems
discussed here, but some features may be useful in studies of qualitative
generic properties in applications to data from real systems which manifestly
seem to exhibit behaviour reminiscent of near-random processes.
