A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential f(x). For rapidly decaying correlations of g, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

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This paper is devotcd to the problem of anomalous transport across ii magnetic ficld that includcs a small stochastic component 68. The pertwhation is assumcd to bc so strongly strclchcd along the background magnctic field U, that the parameter R is large: R h,L,/a >> I (here h, =do,/& << I , and L,, is thc longitudinal and S the tran~ver~e correlation length of the magnetic perturbation). This strong turbulence limit, which is opposite to the quasi-linear one (R << I), has certain notable features. The principal result is that the main transport is conccntrated in very thin regions, being fractal sets with the dimension 4, which can range in value from 2 to 2.75, depending on the spectrum of the magnetic pcrturbation. These regions consist of a small fraction of magnetic lines that pcrcolatc, that is, walk from lhc non-perturbed magnetic flux surfaces to a distance barge compared to the transvcrse correlation length 6. Due to such a strong inhomogeneity of the transport distribution, as well as the long correlations, thc standard transport averaging techniques fdil. and one should make use afthe percolation theory methods. Thus the strong turbulcncc regime is referred 10 here as the percolorion limir. In compilrisoii with the quasilinear limit, the percolation limit has seveial additional intermediate regimes and the cxpressions f i r the efkcectivc hear conductivity xCli include thc critical exponents of 2-D percolation theory. The estimates of zcaarc obtained both in thecollisional andcollisionless limits, includingthccasc ofnon-stationary magnetic perturbations.