A Gaussian distribution for refined DT invariants and 3D partitions

TL;DR

This paper demonstrates that refined Donaldson-Thomas invariants for three-dimensional space follow a Gaussian distribution in the limit, using combinatorial and analytic methods involving 3D partitions and the Hardy-Littlewood circle method.

Contribution

It introduces a new probabilistic understanding of refined DT invariants and extends Wright's asymptotic analysis to the bivariate case using advanced analytic techniques.

Findings

Refined DT invariants exhibit Gaussian distribution in the limit

Weighted counts of 3D partitions are analyzed asymptotically

Application of Hardy-Littlewood circle method to MacMahon's function

Abstract

We show that the refined Donaldson-Thomas invariants of C3, suitably normalized, have a Gaussian distribution as limit law. Combinatorially these numbers are given by weighted counts of 3D partitions. Our technique is to use the Hardy-Littlewood circle method to analyze the bivariate asymptotics of a q-deformation of MacMahon's function. The proof is based on that of E.M. Wright who explored the single variable case.