This talk will introduce the analytic continuation method, a technique for analyzing homogenization in disordered media. Originally built for the study of composites arising in materials science, the technique is based on the spectral properties of a random self-adjoint operator determined by a fixed underlying geometry (that of the homogeneous material) and a random perturbation of the same geometry. The random perturbation is essentially the inhomogeneous material that makes up the composite. The operator is intimately related to the Gaussian Free Field and the Uniform Spanning Tree. Many potential lines of research will be discussed, along with some recent results on the spectral properties that have significantly improved existing numerical methods.
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