The talk is concerned with the rigorous mathematical description of propagation and localisation of waves in a particular class of composite materials with random microscopic geometry, called micro-resonant (or high-contrast) random media: small inclusions of a “soft” material are randomly dispersed in a “stiff” matrix.  The highly contrasting physical properties of the two constituents,  combined with a particular scaling of the inclusions,  result in microscopic resonances, which manifest macroscopically by allowing propagation of waves in the material only within certain ranges of frequencies (band-gap spectrum).

High-contrast media with periodically distributed inclusions have been extensively studied and numerous results are available in the literature.  However,  their stochastic counterparts, which model more realistic scenarios and may exhibit localisation,  are far from being well understood from a mathematical viewpoint.  In my talk I will give an overview of existing results through the prism of stochastic homogenisation and spectral theory,  and discuss recent advances and ongoing work.

Based on joint works with M. Cherdantsev, I. Velčić, P. Bella and M. Täufer.