The GHP scaling limit of uniform spanning trees in high dimensions

We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the \(d\)-dimensional torus \(\mathbb{Z}_n^d\) with \(d>4\), the hypercube \(\{0,1\}^n\), and transitive expander graphs. Several coroll...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Archer, Eleanor, Nachmias, Asaf, Shalev, Matan
Format: Article
Language:English
Subjects:
Diameters
Graph theory
Graphs
Hypercubes
Random walk
Toruses
Online Access:Get full text
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Summary:We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the \(d\)-dimensional torus \(\mathbb{Z}_n^d\) with \(d>4\), the hypercube \(\{0,1\}^n\), and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.
ISSN:2331-8422