Renormalization and finite size correction in extreme statistics

Géza Györgyi

Eötvös University, Institute for Theoretical Physics, Budapest, Hungary

We show a brief and transparent derivation of the finite size correction, that is, amplitude and shape of the correction, for the asymptotic extreme value distributions for iid variables of a large, but finite, number. The traditional limit distributions are in this picture fixed points, whereas the finite size scaling function and the exponent of the size generically correspond to eigenfunctions and eigenvalues, respectively, of the linearized renormalization transformation about a fixed point. Furthermore, a few examples of systems of correlated variables are treated, like subcritical percolation and 1/fα noise, where numerical as well as computer assisted analytical results are presented on finite size corrections. One of our findings is that the correction found in the iid case may be valid also in some classes of correlated variables.