Limits of logarithmic combinatorial structures

Abstract

Under very mild conditions, we prove that the limiting behavior of the component counts in a decomposable logarithmic combinatorial structure conforms to a single, unified pattern, which includes functional central limit theorems, Erdös-Turán laws, Poisson–Dirichlet limits for the large components and Poisson approximation in total variation for the total number ofcomponents. Our approach is entirely probabilistic, and the conditions can readily be verified in practice.