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The diameter of the uniform spanning tree of dense graphs

Part of: Special processes Equilibrium statistical mechanics Graph theory

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: [email protected]
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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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References

Aldous, D. J. (1991) The continuum random tree. I. Ann. Prob. 19(1) 1–28.CrossRefGoogle Scholar
Aldous, D. J. (1991) The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990), pp. 23–70.CrossRefGoogle Scholar
Aldous, D. J. (1993) The continuum random tree. III. Ann. Prob. 21(1) 248–289.CrossRefGoogle Scholar
Alon, N. (1986) Eigenvalues and expanders. Combinatorica 6 83–96.CrossRefGoogle Scholar
Alon, N. and Milman, V. D. (1985) $\lambda 1$ ; isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Ser. B 38(1) 73–88.CrossRefGoogle Scholar
Feder, T. and Mihail, M. (1992) Balanced matroids. In Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, pp. 26–38.CrossRefGoogle Scholar
Jerrum, M. and Sinclair, A. (1989) Approximating the permanent. SIAM J. Comput. 18(6) 1149–1178.CrossRefGoogle Scholar
Jerrum, M., Son, J.-B., Tetali, P. and Vigoda, E. (2004) Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains. Ann. Appl. Prob. 14(4) 1741–1765 (English).CrossRefGoogle Scholar
Kolchin, V. F. (1986) Random Mappings, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York,. Translated from the Russian, With a foreword by S. R. S. Varadhan.Google Scholar
Kwok, T. C., Lau, L. C., Lee, Y. T., Gharan, S. O. and Trevisan, L. (2013) Improved Cheeger’s inequality: analysis of spectral partitioning algorithms through higher order spectral gap. In STOC’13| Proceedings of the 2013 ACM Symposium on Theory of Computing, pp. 11–20.CrossRefGoogle Scholar
Lawler, G. F. and Sokal, A. D. (1988) Bounds on the $L^2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309(2) 557–580.Google Scholar
Le Gall, J.-F. (2005) Random trees and applications. Probab. Surv. 2 245–311.CrossRefGoogle Scholar
Le Gall, J.-F. (2006) Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15(1) 35–62.CrossRefGoogle Scholar
Levin, D. A. and Peres, Y. (2017) Markov Chains and Mixing Times, American Mathematical Society, Providence, RI. Second edition of [ MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson. Available at https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf.Google Scholar
Lyons, R. and Peres, Y. (2016) Probability on Trees and Networks , Vol. 42 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, New York.Google Scholar
Michaeli, P., Nachmias, A. and Shalev, M. (2021) The diameter of uniform spanning trees in high dimensions. Probab. Theory Related Fields 179(1–2) 261–294.CrossRefGoogle Scholar
Peres, Y. and Revelle, D. (2004) Scaling limits of the uniform spanning tree and loop-erased random walk onfinite graphs, arXiv Mathematics e-prints, available at math/0410430.Google Scholar
Szekeres, G. (1983) Distribution of labelled trees by diameter. In Combinatorial Mathematics X (Adelaide, 1982), Lecture Notes in Mathematics, pp. 392–397.CrossRefGoogle Scholar
Wilson, D. B. (1996) Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 296–303.CrossRefGoogle Scholar