The GHP scaling limit of uniform spanning trees in high dimensions

Eleanor Archer; Asaf Nachmias; Matan Shalev

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arXiv · 2112.01203

The GHP scaling limit of uniform spanning trees in high dimensions

Eleanor Archer·Asaf Nachmias·Matan Shalev

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Abstract

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.

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Eleanor Archer, Asaf Nachmias, Matan Shalev. 2021-12-02. The GHP scaling limit of uniform spanning trees in high dimensions. https://arxiv.org/abs/2112.01203

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