Addario-Berry, L and Sen, S (2021) Geometry of the minimal spanning tree of a random 3-regular graph. In: Probability Theory and Related Fields, 180 (3-4). pp. 553-620.
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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 Addario-Berry et al. (2017) 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�s�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. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
| Item Type: | Journal Article |
|---|---|
| Publication: | Probability Theory and Related Fields |
| Publisher: | Springer Science and Business Media Deutschland GmbH |
| Additional Information: | The copyright for this article belongs to Authors |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 30 Aug 2021 06:16 |
| Last Modified: | 30 Aug 2021 06:16 |
| URI: | http://eprints.iisc.ac.in/id/eprint/69555 |
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