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November 2021 Conformal growth rates and spectral geometry on distributional limits of graphs
James R. Lee
Author Affiliations +
Ann. Probab. 49(6): 2671-2731 (November 2021). DOI: 10.1214/20-AOP1480

Abstract

For a unimodular random graph (G,ρ), we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of (G,ρ), which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting consequences for many low-dimensional models.

The consequences in dimension two are particularly strong. It establishes that models like the uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ) almost surely have spectral dimension at most two. It also establishes a conjecture of Benjamini and Schramm (Electron. J. Probab. 6 (2001) no. 23) by extending their recurrence theorem from planar graphs to arbitrary families of H-minor-free graphs. More generally, it strengthens the work of Gurel-Gurevich and Nachmias (Ann. of Math. (2) 177 (2013) 761–781) who established recurrence for distributional limits of planar graphs when the degree of the root has exponential tails.

We further present a general method for proving subdiffusivity of the random walk on a large class of models, including UIPT and UIPQ, using only the volume growth profile of balls in the intrinsic metric.

Acknowledgments

I would like to thank Itai Benjamini for his support and for sharing his many insights, along with a stream of inspiring open questions. My thanks to Asaf Nachmias for reading many drafts of this manuscript and sharing his wisdom on circle packings, and to Omer Angel and Nicolas Curien for invaluable discussions on the geometry of random planar maps. In particular, Nicolas offered crucial guidance for the references and calculations in the Appendix.

I am grateful to Jian Ding, Russ Lyons, Steffen Rohde and Lior Silberman for comments at various stages, and to Tom Hutchcroft who pointed out an error in an initial draft of this manuscript and explained the construction found at the beginning of Section 4.4, as well as a simpler proof of Lemma 4.3. Lastly, I am grateful to the anonymous referees for their detailed, insightful feedback.

Citation

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James R. Lee. "Conformal growth rates and spectral geometry on distributional limits of graphs." Ann. Probab. 49 (6) 2671 - 2731, November 2021. https://doi.org/10.1214/20-AOP1480

Information

Received: 1 July 2020; Published: November 2021
First available in Project Euclid: 7 December 2021

MathSciNet: MR4348678
zbMATH: 1482.05319
Digital Object Identifier: 10.1214/20-AOP1480

Subjects:
Primary: 52C26 , 60D05 , 60J10 , 60-XX

Keywords: Random walk , Spectral dimension , uniform infinite planar triangulation , unimodular random graphs

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 6 • November 2021
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