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

We study the homogenisation problem for elliptic high-contrast operators $\mathcal{A}^\varepsilon$ whose coefficients degenerate as $\varepsilon\to 0$ on a set of randomly distributed inclusions. In our earlier paper [Stochastic homogenisation of high-contrast media. Applicable Analysis (2019)] we proved the Hausdorff convergence of the spectra of $\mathcal{A}^\varepsilon$ to the spectrum of a two-scale limit homogenised operator $\mathcal{A}^{\rm hom}$ in the bounded domain setting and provided a partial analysis of $\mathcal{A}^{\rm hom}$ and its spectrum. In the present work we offer a comprehensive study of their properties. Our main focus, however, is on the spectra of $\mathcal{A}^\varepsilon$ in the whole space setting, when their structure is significantly different to the case of a bounded domain. We show that for the whole space the limit of ${\rm Sp}(\mathcal{A}^\varepsilon)$ is, in general, strictly larger than ${\rm Sp}(\mathcal{A}^{\rm hom})$ and illustrate how this effect is connected with the stochastic nature of the operators in question. Under an additional assumption of finite range correlation of the random inclusions, we are able to characterise the limit $\lim_{\varepsilon\to 0}{\rm Sp}(\mathcal{A}^\varepsilon)$ via a stochastic non-local analogue of Zhikov's $\beta$-function. Furthermore, we introduce the notion of a statistically relevant (limiting) spectrum and develop a qualitative and quantitative tool for describing the part ${\rm Sp}(\mathcal{A}^\varepsilon)$ that converges to ${\rm Sp}(\mathcal{A}^{\rm hom})$.