The Ising model and critical behavior of transport in binary composite media

Abstract

We present a general theory for critical behavior of transport in binary composite media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi–static case (frequency dependent) with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures.