Under very mild conditions, we prove that the limiting behavior of the comp
onent counts in a decomposable logarithmic combinatorial structure conforms
to a single, unified pattern, which includes functional central limit theo
rems, Erdas-Turan laws, Poisson-Dirichlet limits for the large components a
nd Poisson approximation in total variation for the total number of compone
nts. Our approach is entirely probabilistic, and the conditions can readily
be verified in practice.