SELF-SIMILAR POTENTIALS IN RANDOM-MEDIA, FRACTAL EVOLUTIONARY LANDSCAPES AND KIMURAS NEUTRAL THEORY OF MOLECULAR EVOLUTION

A general method for constructing self-similar scalar fractal random f ields is suggested based on the assumption that the fields are generat ed by a broad distribution of punctual sources. The method is illustra ted by a problem of population biology, the evolution on a random fitn ess landscape. A random fitness landscape is constructed based on the following hypotheses. The landscape is defined by the dependence of a fitness variable phi on the state vector x of the individuals: phi = p hi(x); the corresponding hypersurface has a large number of local maxi ma characterized by a local probability law with variable parameters. These maxima are uniformly randomly distributed throughout the state s pace of the individuals. From generation to generation the heights and the shapes of these local maxima can change; this change is described in terms of two probabilities p and alpha that an individual modifica tion occurs and that the process of variation as a whole stops, respec tively. A general method for computing the stochastic properties of th e evolutionary landscape is suggested based on the use of characterist ic functionals. An explicit computation of the Fourier spectrum of the cumulants of the evolutionary landscape is performed in a limit of th e thermodynamic type for which the number of maxima and the volume of the state space of the individuals tend to infinity but the density of maxima remains constant. It is shown that, although a typical realiza tion of the evolutionary landscape is very rough, its average properti es expressed by the Fourier spectrum of its cumulants are smooth and c haracterized by scaling laws of the power law type. The average landsc ape which is made up of the frozen contributions of the changes corres ponding to different generations is flat, a result which is consistent with the Kimura's theory of molecular evolution. Some general implica tions of the suggested approach for the statistical physics of systems with random ultrametric topology are also investigated.