The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

Eleanor Archer; Asaf Nachmias; Matan Shalev

doi:10.1007/s00220-023-04923-2

The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

Eleanor Archer
, Asaf Nachmias
, Matan Shalev

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Article number	73
Journal	Communications in Mathematical Physics
Volume	405
Issue number	3
DOIs	https://doi.org/10.1007/s00220-023-04923-2
State	Published - Mar 2024

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s00220-023-04923-2

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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 Znd 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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N2 - 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 Znd 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.

AB - 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 Znd 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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JO - Communications in Mathematical Physics

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The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

Abstract

ASJC Scopus subject areas

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