A Priori Tests for the MIXMAX Random Number Generator

TL;DR

This paper introduces a priori tests for the MIXMAX pseudo-random number generator, analyzing correlation properties and lattice structure, and finds that MIXMAX exhibits ideal correlation behavior and a stable spectral index, indicating high-quality randomness.

Contribution

The paper develops new a priori tests for linear matrix-recursion generators and demonstrates that MIXMAX has vanishing serial correlations and a robust lattice structure.

Findings

Serial correlation vanishes for MIXMAX.

Lowest non-zero correlator at lag s=1 involves four elements.

Spectral index of MIXMAX is independent of matrix size and equals √3.

Abstract

We define two a priori tests of pseudo-random number generators for the class of linear matrix-recursions. The first desirable property of a random number generator is the smallness of serial or lagged correlations between generated numbers. For the particular matrix generator called MIXMAX, we find that the serial correlation actually vanishes. Next, we define a more sophisticated measure of correlation, which is a multiple correlator between elements of the generated vectors. The lowest order non-vanishing correlator is a four-element correlator and is non-zero for lag $s=1$. At lag $s \ge 2$, this correlator again vanishes. For lag $s=2$, the lowest non-zero correlator is a six-element correlator. The second desirable property for a linear generator is the favorable structure of the lattice which typically appears in dimensions higher than the dimension of the phase space of the generator, as discovered by Marsaglia. We define an appropriate generalization of the notion of the spectral index for LCG which is a measure of goodness of this lattice to the matrix generators such as MIXMAX and find that the spectral index is independent of the size of the matrix N and is equal to $\sqrt{3}$.