Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Peter Nightingale


This dissertation uses the hierarchical q-state Potts model at the critical point to develop a new random number generator test. We start with an exposition of renormalization group approach by means of which one can numerically exactly compute the free energy, specific heat and susceptibility of large, but finite lattices. We then show that generalization of these standard techniques allows one to also compute probability distributions related to the energy and the order parameter. The various computed quantities can be compared with Monte Carlo estimates of the same quantities. We demonstrate that the structure of the hierarchical lattices used allows one to perform the Monte Carlo calculations by direct sampling. This avoids the usual critical slowing down that plagues Monte Carlo calculations at the critical point. As is well known, critical behavior is highly susceptible to perturbations. We expect that flaws of the pseudo random number generator, such as correlations, will cause statistically significant discrepancies between the results of the simulations and the numerically exactly computed results. Details of the computer code generated for these tests are included.