The diameter of the minimum spanning tree of the complete graph with inhomogeneous random weights

We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends on the product of the weights of its endpoints. We then study the minimum spanning tree corresponding to the edge capacities. Under a condition of finite moments on the node weights, we show that the expected diameter and typical distances of this minimum spanning tree are of order $n^{1/3}$. This is a generalization of the results of Addario-Berry, Broutin, and Reed [2009]. We then use our result to answer a conjecture in statistical physics about typical distances on a closely related object. This work also sets the ground for proving the existence of a non-trivial scaling limit of this spanning tree (a generalization of the result of Addario-Berry, Broutin, Goldschmidt, and Miermont [2017])). Our proof is based on a detailed study of rank-1 critical inhomogeneous random graphs, done in Safsafi [2020], and novel couplings between exploration trees related to those graphs and Galton-Watson trees.