Percolation theory and conductivity of strongly inhomogeneous media

Percolation theory is the name given a new mathematical discipline, initiated in 1957 and found to be unusually productive in the physics of inhomogeneous systems. The main purpose of the review is to describe the recently rapidly developing theory of the electric conductivity of strongly inhomogeneous media, which leads to questions of percolation theory. This gives rise to new problems in percolation theory, namely continual problems and problems dealing with random sites. The methods of solving these problems and the main results are described in detail. Particular attention is paid to the behavior of various quantities near the percolation threshold, and the theory of critical exponents and the similarity hypothesis in percolation theory are treated in detail for the first time. The main objects to which the theory is applied are amorphous and weakly-doped crystalline semiconductors. The main results of the theory of hopping conductivity of such semiconductors are reported, namely, the exponential temperature and concentration dependences, the theory of magnetoresistance, and the theory of the conductivity of films. The structure of the pre-exponential terms is also discussed.