A Meinardus theorem with multiple singularities

TL;DR

This paper extends Meinardus' theorem to cases with multiple singularities, providing new asymptotic formulas for combinatorial structures with Dirichlet generating functions having multiple poles, challenging previous assumptions in physics literature.

Contribution

It generalizes Meinardus' theorem to multiple poles, offering new asymptotic results for decomposable structures with applications in physics and combinatorics.

Findings

Derived asymptotics for structures with multiple poles

Disproved the belief that the rightmost pole dominates asymptotics

Applicable to vector partitions and quantum field theory

Abstract

Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size $n$, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.