Central Limit Theorems via Analytic Combinatorics in Several Variables

Abstract

The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of analytic combinatorics in several variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. In this work we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving combinatorial limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large variety of combinatorial generating functions. To illustrate the techniques involved, we also establish explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem stated as a conjecture in recent work of Chung et al.), integer compositions, and $n$-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate $LU$-factorization. It is our hope that this work provides readers a blueprint to apply the powerful tools of ACSV to prove central limit theorems in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields.