A Gaussian distribution for refined DT invariants and 3D partitions

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.