Ahlfors regular conformal dimension of metrics on infinite graphs and spectral dimension of the associated random walks

Abstract

Quasisymmetry is a well-studied property of homeomorphisms between metric spaces, and the 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.