  We develop a qualitative homogenisation and spectral theory for elliptic
systems of partial differential equations in divergence form with highly
contrasting (i.e., non uniformly elliptic) random coefficients. The focus of
the paper is on the behaviour of the spectrum as the heterogeneity parameter
tends to zero; in particular, we show that in general one doesn't have
Hausdorff convergence of spectra. The theoretical analysis is complemented by
several explicit examples, showcasing the wider range of applications and
physical effects of systems with random coefficients, when compared with
systems with periodic coefficients or with scalar operators (both random and
periodic).
