Some relation between spectral dimension and Ahlfors regular conformal dimension on infinite graphs

Abstract

The spectral dimension $d_s$ of a weighted graph is an exponent associated with the asymptotic behavior of the random walk on the graph. The Ahlfors regular conformal dimension ${\dim }_{\mathrm{ARC}}$ of the graph distance is a quasisymmetric invariant, where quasisymmetry is a well-studied property of homeomorphisms between metric spaces. In this paper, we give a typical example of a fractal-like graph with $d_s<{\dim }_{\mathrm{ARC}}<2$ and prove a sufficient condition for ${\dim }_{\mathrm{ARC}}\le d_s<2.$

27 pages, 13 figures. This article was revised to prepare for submission to a journal. In particular, the order of sections and statements was changed. Theorem 3.2 and Proposition 3.11 of the first version, which are cited in arXiv:2211.11473, were moved to Theorem 2.2 and Proposition 4.1, respectively

References in corpus (2)

• Systems of Dyadic Cubes of Complete, Doubling, Uniformly Perfect Metric Spaces without Detours
• Ahlfors Regular Conformal Dimension of Metrics on Infinite Graphs and Spectral Dimension of the Associated Random Walks