A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures

Abstract

A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines conditions under which the total variation distance between the true joint distribution and the approximation should be small, and was successfully exploited by Pittel in the cases of integer partitions and set partitions. We provide sufficient conditions for this principle to be true in a general context, valid for certain discrete probability distributions which are $\textit{perturbed log-concave}$, via a quantitative local central limit theorem. We then use it to generalize some classical asymptotic statistics in combinatorial theory, as well as assert some new ones.