Ahlfors Regular Conformal Dimension of Metrics on Infinite Graphs and Spectral Dimension of the Associated Random Walks

TL;DR

This paper explores the relationship between Ahlfors regular conformal dimension, spectral dimension, and critical exponents of p-energies on infinite graphs, revealing their interconnectedness.

Contribution

It establishes that the Ahlfors regular conformal dimension coincides with the critical exponent of p-energies and relates it to the spectral dimension of the graph.

Findings

Ahlfors regular conformal dimension equals the critical exponent of p-energies.

A relation between conformal dimension and spectral dimension is demonstrated.

Provides a new perspective on the invariants of infinite graphs.

Abstract

Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and Ahlfors regular conformal dimension is a quasisymmetric invariant. In the present paper, we consider the Ahlfors regular conformal dimension of metrics on infinite graphs, and show that this notion coincides with the critical exponent of $p$-energies. Moreover, we give a relation between the Ahlfors regular conformal dimension and the spectral dimension of a graph.