Testing Randomness by Means of Random Matrix Theory

Random matrix theory (RMT) derives, at the limit of both the dimension N and the length of sequences L going to infinity, that the eigenvalue distribution of the cross correlation matrix with high random nature can be expressed by one function of Q = L/N .U sing this fact, we propose a new method of testing randomness of a given sequence. Namely, a sequence passes the test if the eigenvalue distribution of the cross correlation matrix made of the pieces of a given sequence matches the corresponding theoretical curve derived by RMT, and fails otherwise. The comparison is quantified by employing the moments of the eigenvalue distribution to its theoretical counterparts. We have tested its performance on five kinds of test data including the Linear Congruential Generator (LCG), the Mersenne Twister (MT), and three physical random number generators, and confirmed that all the five pass the test. However, the method can distinguish the difference of randomness of the derivatives of random sequences, and the initial part of LCG, which are distinctly less random than the original sequences.