The Art of Walking on Fractal Spaces and Random Media

Abstract

Since we know that mountain shapes are natural fractals, the title would be appropriate for a lecture on climbing the Mont Blanc, the Aiguille Verte or any of the peaks we can contemplate from Les Houches. My subject is more down to earth and only consists in a short review of recent work on different types of walks on fractal lattices and on random systems such as percolation clusters. The first part is a reminder of geometrical and physical aspects of dimensionality, and discusses why several dimensions are needed to characterize fractal objects. The second part is devoted to “simple” walks, mainly random walks, and gives a short account of classical diffusion and conduction in random media. Several reviews of these aspects have appeared recently, by ALEXANDER |1|, MITESCU and ROUSSENQ |2|, RAMMAL |3| and AHARONY |4|, and the reader is referred to these papers for deeper discussions and for technical points. The following parts deal with “advanced” walks, where additional rules are specified: this leads to a large variety of problems and to a wealth of new physical effects. The examples include random walks in the presence of traps, biased walks (e.g., dispersion of particles in a flow through a porous medium), selfavoiding walks. For all these problems, the lack of translational invariance of the lattice (fractal or random) introduces new features, and even the simplest situations may provide surprises.