We propose three physical tests to measure correlations in random numbers used in Monte Carlo simulations. The rst test uses autocorrelation times of certain physical quantities when the Ising model is simulated with the Wol algorithm. The second test is based on random walks, and the third on blocks of n successive numbers. We apply the tests to show that recent errors in high precision simulations using generalized feedback shift register algorithms are due to short range correlations in random number sequences. We also determine the length of these correlations.

We present and analyze in detail a test bench for random number sequences based on the use of physical models. The rst two tests, namely the cluster test and the autocorrelation test, are based on exactly known properties of the two{dimensional Ising model. The other two, the random walk test and the n{block test, are based on random walks on lattices. We have applied these tests to a number of commonly used pseudorandom number generators. The cluster test is shown to be particularly e cient in detecting periodic correlations on bit level, while the autocorrelation, the random walk, and the n{block tests are very well suited for studies of weak correlations in random number sequences. Based on the test results, we demonstrate the reasons behind errors in recent high precision Monte Carlo simulations, and discuss how these could be avoided.