  We investigate and contrast, via entropic sampling based on the Wang-Landau
algorithm, the effects of quenched bond randomness on the critical behavior of
two Ising spin models in 2D. The random bond version of the
superantiferromagnetic (SAF) square model with nearest- and
next-nearest-neighbor competing interactions and the corresponding version of
the simple Ising model are studied and their general universality aspects are
inspected by a detailed finite-size scaling (FSS) analysis. We find that, the
random bond SAF model obeys weak universality, hyperscaling, and exhibits a
strong saturating behavior of the specific heat due to the competing nature of
interactions. On the other hand, for the random Ising model we encounter some
difficulties for a definite discrimination between the two well-known scenarios
of the logarithmic corrections versus the weak universality. Yet, a careful FSS
analysis of our data favors the field-theoretically predicted logarithmic
corrections.
