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We are interested in the properties of multipartite entanglement of a systemcomposed by $n$ $d$-level parties (qudits).  Focussing our attention on pure states we want to tackle the problem of themaximization of the entanglement for such systems. In particular we effort theproblem trying to minimize the purity of the system. It has been shown that notfor all systems this function can reach its lower bound, however it can beproved that for all values of $n$ a $d$ can always be found such that the lowerbound can be reached.  In this paper we examine the high-temperature expansion of the distributionfunction of the bipartite purity over all balanced bipartition considering itsoptimization problem as a problem of statistical mechanics. In particular weprove that the series characterizing the expansion converges and we analyze thebehavior of each term of the series as $d\to \infty$.

Feynman graphs and the large dimensional limit of multipartite entanglement