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

Abstract

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 (Addario-Berry et al. in Ann Probab 45(5):3075-3144, 2017). 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 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 Erdős-Rényi 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.

Stable graphs: distributions and line-breaking construction

“…as n → ∞, in the Gromov-Hausdorff-Prokhorov sense, where the limit space (M, d, µ) is a random measured R-tree having Minkowski dimension 3 almost surely. This convergence has, up to a constant factor, recently been shown by Addario-Berry and Sen [ABS21] to hold also for the MST of a uniform random 3-regular (simple) graph or for the MST of a 3-regular configuration model.…”

On breadth‐first constructions of scaling limits of random graphs and random unicellular maps

“…The space is simply —the Brownian continuum random tree [7, 9]. For , () can be proved by arguments similar to the ones used in [3, 14]; a brief sketch of the proof is given in [6, Section A.1].…”

“…The scaling limit of the minimal spanning tree of the complete graph identified in [4] can be expressed as the limit, as , of the space obtained by applying a “cycle‐breaking” procedure on the space [6, Theorem 4.8]. We now describe one construction of .…”

“…The spaces , , are the building blocks for the scaling limit of the critical Erdős–Rényi random graph identified in [3]; see [6, Construction 3.10 and Theorem 3.12] for a precise statement. The Erdős–Rényi scaling limit is a universal object in the sense that it arises as the scaling limit of a wide array of standard models of critical random discrete structures exhibiting mean‐field behavior.…”