Random Geometric Series
Statistical Mechanics
2007-05-23 v1 Disordered Systems and Neural Networks
Probability
Abstract
Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, <x_n^s> n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.
Cite
@article{arxiv.cond-mat/0403157,
title = {Random Geometric Series},
author = {E. Ben-Naim and P. L. Krapivsky},
journal= {arXiv preprint arXiv:cond-mat/0403157},
year = {2007}
}
Comments
6 pages, 2 figures