Some exact properties of hopping conduction

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Abstract
The ac conductivity of carriers which perform a random walk in a random environment is studied. Irreducible clusters of hopping sites are introduced by connecting sites which share a nonzero jump rate. A finite irreducible random-walk matrix is shown to be negative semidefinite and have a nondegenerate zero eigenvalue. As a result, the real and imaginary parts of the ac conductivity of systems without infinite irreducible clusters vanish, respectively, quadratically and linearly with the frequency. The coefficient of the linear dependence of the imaginary part of the ac conductivity is related to a correlation length of the irreducible clusters. The coefficient for percolation models in a square lattice is obtained by a computer simulation and compared with the result given by the coherent medium approximation. With the use of the dc conductivity and localization criteria in a strong and weak sense, three percolation thresholds besides the usual one are defined, and relations among these thresholds are discussed.

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