Self-similar potentials in random media, fractal evolutionary landscapes and Kimuraʹs neutral theory of molecular evolution

A general method for constructing self-similar scalar fractal random fields is suggested based on the assumption that the fields are generated by a broad distribution of punctual sources. The method is illustrated by a problem of population biology, the evolution on a random fitness landscape. A random fitness landscape is constructed based on the following hypotheses. The landscape is defined by the dependence of a fitness variable φ on the state vector x of the individuals: φ = φ(x); the corresponding hypersurface has a large number of local maxima characterized by a local probability law with variable parameters. These maxima are uniformly randomly distributed throughout the state space 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 α that an individual modification occurs and that the process of variation as a whole stops, respectively. A general method for computing the stochastic properties of the evolutionary landscape is suggested based on the use of characteristic functionals. An explicit computation of the Fourier spectrum of the cumulants of the evolutionary landscape is performed in a limit of the 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 realization of the evolutionary landscape is very rough, its average properties expressed by the Fourier spectrum of its cumulants are smooth and characterized by scaling laws of the power law type. The average landscape which is made up of the frozen contributions of the changes corresponding to different generations is flat, a result which is consistent with the Kimuraʹs theory of molecular evolution. Some general implications of the suggested approach for the statistical physics of systems with random ultrametric topology are also investigated.