50g: an interesting RAND anomaly

[ttw]

RNG failures can be obvious or subtle. I'll describe two cases I've seen in the literature. First, one simulation was with a 45-bit Lehmer generator. Good enough for small stuff. In this case, simulations were being done on a 512x512x512 cube. Using 1 RNG per site, we see that one has 27 repeated bits per cycle so that there are 2^18 rather than 2^45 or larger number of states being sampled. I would claim that the investigator should have known about this ahead of time; he did take precautions long before publication though.

A second and more subtle case came with a 31-bit generator on a 10001x10001 simulation. Note that this system has 2^31*10001^2 different states in the system. Then the investigator took line averages giving 10001 different results. (These were spatial averages.) In the final part of the problem, he took differences of these averages (A1-A2, A2-A3, A3-A4; A1-A3, A2-A4....). Note that one can average these differences to improve accuracy even though one isn't permitted to use these averages to improve variance computations. What happens is that he saw a "signal" that increased with spacings of powers of 2. At 8192 (the biggest), the cross stream (x1,x2,x3...) generators look good but the down stream becomes (x1-x8193, x2-x8194...). The random numbers share their 13 lower bits. Whatever the guy did detected this. Rather subtle trap.