Convergence Laws for Extensions of First-Order Logic with Averaging

For many standard models of random structure, first-order logic sentences exhibit a convergence phenomenon on random inputs. The most well-known example is for random graphs with constant edge probability, where the probabilities of first-order sentences converge to 0 or 1. In other cases, such as certain ``sparse random graph'' models, the probabilities of sentences converge, although not necessarily to 0 or 1. In this work we deal with extensions of first-order logic with aggregate operators, variations of averaging. These logics will consist of real-valued terms, and we allow arbitrary Lipschitz functions to be used as ``connectives''. We show that some of the well-known convergence laws extend to this setting.