High-contrast random composites: homogenisation framework and new spectral phenomena

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. By combining stochastic techniques with operator theory, we link objects defined on an abstract probability space to their counterparts in the physical space. In particular, we show that the key characteristics of the limit operator can be obtained from as single “typical” realisation in the physical space. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of statistically relevant limiting spectrum, we describe the connection between these two sets.