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

TL;DR

This paper develops a multiscale analysis framework for high-contrast random elliptic operators, revealing new spectral phenomena and differences from periodic cases through detailed spectral analysis and the introduction of a novel limiting spectrum concept.

Contribution

It introduces a homogenisation framework for high-contrast random composites and characterizes the spectral limits, highlighting differences from periodic structures.

Findings

Spectral limits differ from the homogenised operator in the random setting.

A new notion of limiting spectrum is introduced.

Full characterization of the spectral convergence under certain random configurations.

Abstract

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. 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 the relevant limiting spectrum, we describe the connection between these two sets.