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

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.