# Review of High-Quality Random Number Generators

## TL;DR

This review discusses high-quality pseudorandom number generators based on the Kolmogorov-Anosov mixing theory, highlighting RANLUX, RANLUX++, and MIXMAX as suitable for demanding Monte Carlo simulations.

## Contribution

It establishes criteria based on mixing theory to evaluate RNG quality and reviews generators that meet these standards, including RANLUX, RANLUX++, and adaptable MIXMAX variants.

## Key findings

- RANLUX and RANLUX++ meet the high-quality criteria.
- Mixmax can be modified to meet the criteria.
- Mixing theory provides a rigorous basis for RNG quality assessment.

## Abstract

This is a review of pseudorandom number generators (RNG's) of the highest quality, suitable for use in the most demanding Monte Carlo calculations. All the RNG's we recommend here are based on the Kolmogorov-Anosov theory of mixing in classical mechanical systems, which guarantees under certain conditions and in certain asymptotic limits, that points on the trajectories of these systems can be used to produce random number sequences of exceptional quality. We outline this theory of mixing and establish criteria for deciding which RNG's are sufficiently good approximations to the ideal mathematical systems that guarantee highest quality. The well-known RANLUX (at highest luxury level) and its recent variant RANLUX++ are seen to meet our criteria, and some of the proposed versions of MIXMAX can be modified easily to meet the same criteria.