English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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

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