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The diameter of the uniform spanning tree of dense graphs

Part of: Equilibrium statistical mechanics Graph theory Special processes

Published online by Cambridge University Press:  13 May 2022

Noga Alon ,
Asaf Nachmias  and
Matan Shalev
Noga Alon
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Asaf Nachmias
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Matan Shalev*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
*
*Corresponding author. Email: matanshalev@mail.tau.ac.il
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Abstract

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on n vertices with minimal degree linear in n is typically of order $\sqrt{n}$ . A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.

Keywords

Uniform spanning treesdense graphsspectral gap

MSC classification

Primary: 05C81: Random walks on graphs 82B41: Random walks, random surfaces, lattice animals, etc.
Secondary: 05C05: Trees 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 6 , November 2022 , pp. 1010 - 1030
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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