© 2018 Author(s). In this paper, we extend the analysis of multipartite entanglement, based on techniques from classical statistical mechanics, to a system composed of n d-level parties (qudits). We introduce a suitable partition function at a fictitious temperature with the average local purity of the system as Hamiltonian. In particular, we analyze the high-temperature expansion of this partition function, prove the convergence of the series, and study its asymptotic behavior as d → ∞. We make use of a diagrammatic technique, classify the graphs, and study their degeneracy. We are thus able to evaluate their contributions and estimate the moments of the distribution of the local purity.