Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs

TL;DR

This paper investigates the heat kernel, jump kernel, and Dirichlet forms on ultrametric Cantor sets derived from higher-rank graphs, revealing their geometric and spectral properties.

Contribution

It introduces a new Dirichlet form linked to spectral triples on ultrametric Cantor sets from higher-rank graphs, analyzing heat kernels and jump processes.

Findings

Volume doubling property of the Perron-Frobenius measure

Asymptotic behavior of the heat kernel

Equivalence of Dirichlet forms and jump kernels

Abstract

The aim of this paper is to study the heat kernel and jump kernel of the Dirichlet form associated to ultrametric Cantor sets $\partial\BB_\Lambda$ that is the infinite path space of the stationary $k$-Bratteli diagram $\BB_\Lambda$, where $\Lambda$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\PB_\Lambda), \pi_\phi, \mathcal{H}, D, \Gamma)$ and is given by \[ Q_s(f,g)=\frac{1}{2} \int_{\Xi} \operatorname{Tr}\big(\vert D\vert^{-s} [D,\pi_{\phi}(f)]^{\ast} [D,\pi_\phi(g)] \big) \, d\nu(\phi), \] where $\Xi$ is the space of choice functions on $\partial \BB_\Lambda \times \partial \BB_\Lambda$. There are two ultrametrics, $d^{(s)}$ and $d_{w_\delta}$, on $\partial \BB_\Lambda$ which make the infinite path space $\PB_\Lambda$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of Laplace-Beltrami operator $\Delta_s$ associated to $Q_s$, and the latter $d_{w_\delta}$ is associated to a weight function $w_\delta$ on $\BB_\Lambda$, where $\delta\in (0,1)$. We show that the Perron-Frobenius measure $\mu$ on $\partial \BB_\Lambda$ has the volume doubling property with respect to both $d^{(s)}$ and $d_{w_\delta}$ and we study the asymptotic behaviors of the heat kernel associated to $Q_s$. Moreover, we show that the Dirichlet form $Q_s$ coincides with a Dirichlet form $\mathcal{Q}_{J_s, \mu}$ which is associated to a jump kernel $J_s$ and the measure $\mu$ on $\partial \BB_\Lambda$, and we investigate the asymptotic behavior and moments of displacements of the process.