Modular Transformations, Order-Chaos Transitions and Pseudo-Random Number Generation

Abstract

Successive pairs of pseudo-random numbers generated by standard linear congruential transformations display ordered patterns of parallel lines. We study the ``ordered'' and ``chaotic'' distribution of such pairs by solving the eigenvalue problem for two-dimensional modular transformations over integers. We conjecture that the optimal uniformity for pair distribution is obtained when the slope of linear modular eigenspaces takes the value $n_{opt}=maxint(p/\sqrt{p-1})$, where $p$ is a prime number. We then propose a new generator of pairs of independent pseudo-random numbers, which realizes an optimal uniform distribution (in the ``statistical'' sense) of points on the unit square $(0,1]\times (0,1]$. The method can be easily generalized to the generation of $k$-tuples of random numbers (with $k>2$)

13 pages, Revtex - 7 Figs - Int. J. Mod. Phys. C, in press

References in corpus (1)

• New Universal Aspects of Diffusion in Strongly Chaotic Systems

Cited by in corpus (3)

• Periodic orbits of the ensemble of Sinai-Arnold cat maps and pseudorandom number generation
• An Introduction to Monte Carlo Simulation of Statistical physics Problem
• Monte Carlo: Basics