FINITE-SIZE-SCALING FOR CELLULAR-AUTOMATA WITH RANDOMIZED GRIDS AND FOR FRACTAL RANDOM-FIELDS IN DISORDERED-SYSTEMS

The finite-size effect on the statistics of independent random point p rocesses is analyzed in connection with the theory of cellular automat a with randomized grids. The space distribution of a large but finite number N-0 of independent particles confined in a large region of d(s) -dimensional Euclidean space of size V-Sigma is investigated by using the technique of characteristic functionals. Exact formal expressions are derived for all many-body correlation functions of the positions o f the particles as well as for all cumulants of the concentration held . These functions are made up of the contributions of the different ne gative powers of the total number N-0 of particles from the system Sig ma(m=0)(n-1)A(n)(m)(N-0)(-m) where n = 1, 2,... are the orders of the correlation functions or of the cumulants of the concentration field. In the thermodynamic limit N-0, V-Sigma --> infinity with N-0/V-Sigma = constant only the terms A(n)(0) survive; the other terms A(n)(m) wit h m > 0 express the finite-size effects. It is shown that, even though the particles are independent, for a finite size of the system a corr elation effect different from zero exists among their positions and th is correlation vanishes in the limit of an infinite size. The correlat ion among the positions of the different particles is a finite-size ef fect due to the conservation of the total number of particles which is similar to the correlation among ideal bosons or fermions at low abso lute temperatures. The stochastic properties of an additive scalar fie ld generated by a random distribution of independent particles are inv estigated. The approach can be applied to the study of stochastic grav itational fluctuations generated by a random distribution of stars or galaxies, of the short-range mean field generated by the particles mak ing up a disordered medium, or of the distribution of the offspring nu mber generated by a plant population randomly distributed in space. Sp ecial attention is paid to the finite-size scaling corrections to the long-range self-similar fractal fields. The computations lead to the s urprising result that for large values of the resulting field the fini te size of the system has practically no influence on the tails of the probability densities of the resulting held, which obey a statistical fractal scaling law of the negative power law type. This apparent par adoxical effect is due to the fact that the very large values of the h eld corresponding to the tails of the probability densities are genera ted by the closest neighbor of the test particle considered and are no t influenced by the more distant particles.