Phystca D 38 f 1989) 270-286 North-Holland. Amsterdam THEORETICAL CONCEPTS FOR FRACTAL G ROWTH L. PIETRONERO Dtparttmento dt Fisica. Unit'ersttd dt Roma "'La Sapwnza". Pm:-ale.4. Moro 2. 0018~ Rome. Italy After the introduction of flactal geometry by Benoit Mandelbrot the ke~ probl*.m is to understand why nature gt~es nsc to fraetal structures. This tmplies the formulation of models of fractal growth based on physical phenomena and the subseq' -m understanding of their mathematical structure in the ~me sense as the renormalization group has allowed to ,,nderstand ,,ng- type models. The models of diffusion-limited aggregatton and the more general dtelecttac breakdown model, based on t,,ranve processes governed by the Laplace equation and a stochastic field, have a clear phystcal meanmg and the~ spontaneously evobe into random fractal structures of gt~at complexity. From a theoretical point of view however it is not possible to describe them within usual concepts. Recently we have introduced a new theo,etical framework for this class of problems. This clarifies the origin of fractal structures in these models and provides a systematic m,'thod for the calculation of the fractai dtmension and the multifractal properties. Here ~,e summarize the basic ideas of this new approach and report about recent developments. I. Introduction "'Fractal geometry is one of those concepts which at first sight inx ires disbelief but on a second thought becomes so natural that one wonders why it has onl.~ recently been dc~eloped". These words by Berry [ I 1 from his review of Mandelbrot's book [2] explain why fractal geometry is having an important influ- ence m all scientific disciplines and in particular in physics. This concept was clearly lacking for the de- scription of complex structures in nature and, by in- troducing it, Mandelbrot has provided a playground of new problems concerning basic properties of nat- ural phenomena. These problems were up to now left at the margins ofscientific activity because it was not possible to cast them within the framework of math- ematical methods based on analyticity. From the point ofview of fractal geometry it is now possible to pose these problems correctly. Thts fact, by itself, has far reaching consequences also on problems that have been object of extensive study along traditional lines. An interesting example is the problem of the statistical properties of the large scale dtstribution of matter in the universe. Since a few years the complete three-dimensional distribu- Essa~ m honour of Bcnoit B Mandelbrot Fractals in Phystcs- A. Aharon~ and J. Feder ledttors) tion ofgalaxies in a certain luminosity range is avail- able for appreciably large volumes. It is possible therefore to perform statistical analysis of these dis- tributions [3]. This has been done extenstveb usirg mathematical methods that assume a priori homo- geneity at large scale. The reasons for this assump- tion are basicalb historical and due to the following argument. The cosmological principle implies local isotropy, this together w!lh analyttcttx leads to ho. mogeneity [4]. In the absence ofaP alternative the- oretical framework, analyticity was not considered as a property to be checked from experimental data but it was somehow included in the cosmological princi- ple itself. Fractal geometry makes clear instead that local isotropy is not necessarib associated with ho- mogeneity [2]. It was natural therefore to reconsider the statistical analysis of galaxy distribution from a more general point of ~iew that does not assume ho- mogeneily a priori. This has given rise to a surprising result [5 ]. Contrary to the previous conclustons[ 3 ]. the galaxy distribution does not sl-.a~v ar,~ tendc~c.~ to homogenize, so the impliot assumption of ana- lyticity is actually incorrect. This new analysts poiots therefore to the possibility that the large scale distri- bution of matter in the inverse is fractal to al! ob- 01 b7.278Q/8q/$0.L50,t_-", Elves !cr Sctence Purqtshcrs B.x' ( North-Holland Physics Pubhshmg 'Oo,tston )