Abstract
For reversible random networks, we exhibit a relationship between the almost sure spectral dimension and the Euclidean growth exponent, which is the smallest asymptotic rate of volume growth over all embeddings of the network into a Hilbert space. Using metric embedding theory, it is then shown that the Euclidean growth exponent coincides with the metric growth exponent. This simplifies and generalizes a powerful tool for bounding the spectral dimension in random networks.
This is a preview of subscription content, log in via an institution to check access.
References
O. Angel, T. Hutchcroft, A. Nachmias and G. Ray, Hyperbolic and parabolic unimodular random maps, Geometric and Functional Analysis 28 (2018), 879–942.
D. Aldous and R. Lyons, Processes on unimodular random networks, Electronic Journal of Probability 12 (2007), 1454–1508.
O. Angel and O. Schramm, Uniform infinite planar triangulations, Communications in Mathematical Physics 241 (2003), 191–213.
K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric and Functional Analysis 2 (1992), 137–172.
I. Benjamini and N. Curien, Ergodic theory on stationary random graphs, Electronic Journal of Probability 17 (2012), Article no. 2401.
J. Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel Journal of Mathematics 52 (1985), 46–52.
I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs, Electronic Journal of Probability 6 (2001), Article no. 96.
E. Gwynne and T. Hutchcroft, Anomalous diffusion of random walk on random planar maps, Probability Theory and Related Fields 178 (2020), 567–611.
S. Ganguly and J. R. Lee, Chemical subdiffusivity of critical 2D percolation, Communications in Mathematical Physics 389 (2022), 695–714.
S. Ganguly, J. R. Lee and Y. Peres, Diffusive estimates for random walks on stationary random graphs of polynomial growth, Geometric and Functional Analysis 27 (2017), 596–630.
E. Gwynne and J. Miller, Random walk on random planar maps: Spectral dimension, resistance and displacement, Annals of Probability 49 (2021), 1097–1128.
M. Krikun, Local structure of random quadrangulations, https://arXiv.org/abs/math/0512304.
J. R. Lee, Discrete uniformizing metrics on distributional limits of sphere packings, Geometric and Functional Analysis 28 (2018), 1091–1130.
J. R. Lee, Conformal growth rates and spectral geometry on distributional limits of graphs, Annals of Probability 49 (2021), 2671–2731.
J. R. Lee, Relations between scaling exponents in unimodular random graphs, https://arxiv.org/abs/2007.06548.
N. Linial, E. London and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications, Combinatorica 15 (1995), 215–245.
N. Linial, A. Magen and A. Naor, Girth and Euclidean distortion, Geometric and Functional Analysis 12 (2002), 380–394.
A. Naor, Y. Peres, O. Schramm and S. Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Mathematical Journal 134 (2006), 165–197.
Author information
Authors and Affiliations
Corresponding author
Additional information
In honor of Nati Linial on the occasion of his 70th birthday
Rights and permissions
About this article
Cite this article
Lee, J.R. Spectral dimension, Euclidean embeddings, and the metric growth exponent. Isr. J. Math. 256, 417–439 (2023). https://doi.org/10.1007/s11856-023-2520-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2520-x