Geometry of the minimal spanning tree of a random $3$-regular graph

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph [5]. In this work, we show that the MST constructed by assigning i.i.d. continuous edge-weights to either the random (simple) $3$-regular graph or the $3$-regular configuration model on $n$ vertices, endowed with the tree distance scaled by $n^{-1/3}$ and the uniform probability measure on the vertices, converges in distribution with respect to Gromov-Hausdorff-Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in [5] up to a scaling factor of $6^{1/3}$. Our proof relies on a novel argument that proceeds via a comparison between a $3$-regular configuration model and the largest component in the critical Erd\H{o}s-R\'enyi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.